 
 
 
 
 
 
 
Dynamic Differential Geometry  
in an Educational Augmented 
Reality Application 
 
 
DIPLOMARBEIT 
 
zur Erlangung des akademischen Grades 
 
Diplom-Ingenieurin 
 
im Rahmen des Studiums 
 
Medieninformatik 
 
eingereicht von 
 
Marie-Theres Tschurlovits 
Matrikelnummer 0125975 
 
 
an der 
Fakultät für Informatik der Technischen Universität Wien  
 
 
 
Betreuer: Univ. Ass. Mag. Dr. Hannes Kaufmann 
 
 
 
 
Wien, 20.08.2008          ____________________                ____________________ 
        (Unterschrift Verfasserin)             (Unterschrift Betreuer) 
 
 
 
 
Technische Universität Wien 
A-1040 Wien ▪ Karlsplatz 13 ▪ Tel. +43/(0)1/58801-0 ▪ http://www.tuwien.ac.at 
 
 
Die approbierte Originalversion dieser Diplom-/Masterarbeit ist an der 
Hauptbibliothek der Technischen Universität Wien aufgestellt  
(http://www.ub.tuwien.ac.at). 
 
The approved original version of this diploma or master thesis is available at the 
main library of the Vienna University of Technology   
(http://www.ub.tuwien.ac.at/englweb/). 
 
Abstract
In this thesis a number of geometry software packages leading both to static
and dynamic constructions and their particular features will be presented. Af-
terwards Construct3D [14] a 3D dynamic geometry construction tool will be
introduced. It is based on the Augmented Reality System Studierstube. Con-
struct3D's greatest advantage compared to other dynamic geometry software
is the possibility for users to see the real environment augmented with virtual
content with the aid of a head mounted display. That gives the users, mainly
high school and university students, the opportunity to actually construct,
explore and interact with three dimensional content in "real" 3D space.
The practical part of this thesis was the implementation of a number of new
functions for Construct3D. Several tools have been developed to enhance the
understanding of the term curvature of curves and surfaces. To complement
the already available sweep function of Construct3D helical and general sweeps
have been implemented.
2
Kurzfassung
In dieser Arbeit wird eine Reihe von Geometrie Software Paketen präsentiert,
die sowohl zu statischen als auch zu dynamischen Konstruktionen führen. An-
schlieÿend wird Construct3D eingeführt, eine Software zu Erstellung dyna-
mischer 3D Konstruktionen, die auf dem Augmented Reality System Studier-
stube basiert. Der gröÿte Vorteil von Construct3D gegenüber anderen dy-
namischen Geometrie Software Paketen ist die Option mit Hilfe eines Head
mounted displays die echte Umgebung angereichert mit virtuellen Inhalten be-
trachten zu können. Das erönet den Usern, vornehmlich Oberstufenschülern
und Studenten, die Möglichkeit im "richtigen" dreidimensionalen Raum dreidi-
mensionale Objekte zu konstruieren, zu erforschen und in weiterer Folge damit
zu interagieren.
Der praktische Teil dieser Arbeit umfasste die Implementierung einer Reihe
von neuen Funktionen für Construct3D. Es wurden einige Werkzeuge entwick-
elt um das Verständnis des Begris der Krümmung von Kurven und Flächen zu
fördern. Um die bereits vorhandenen Sweep Funktionen zu ergänzen wurden
auÿerdem Schraub- und Schiebeächen implementiert.
3
Acknowledgements
First of all I would like to thank my supervisor Hannes Kaufmann for his
comprehensive support. He supported me in every problem or question I had,
also at night, weekends and during holidays. His colleague Mathis Csisinko
helped me when problems with Construct3D occured.
I am also grateful to Marion Papp who supported me with geometrical back-
ground informations and was one of the rst testers of my work.
My partner Thomas Weiÿ advised me in programming questions and was
the man for mental support.
Last but not least I want to thank my parents who gave me the opportunity
to study. I know my father is very proud I studied at the university where he
has worked himself for so many years.
4
Contents
Contents
1 Motivation 8
2 Related Work 9
2.1 History of dynamic geometry software . . . . . . . . . . . . . . . 9
2.2 Dynamic geometry software . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Dynamic 2D Geometry software . . . . . . . . . . . . . . 10
2.2.2 Dynamic 3D Geometry software . . . . . . . . . . . . . . 13
2.2.3 CAD software providing dierential geometry functionality 15
3 Technological Foundations 18
3.1 Open Inventor and Coin3D . . . . . . . . . . . . . . . . . . . . . 18
3.1.1 The Open Inventor Toolkit . . . . . . . . . . . . . . . . . 18
3.1.2 The Scene Database . . . . . . . . . . . . . . . . . . . . 19
3.1.3 Node Kits . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Studierstube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 3D event system . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Widget system . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.3 Dynamic application loading . . . . . . . . . . . . . . . . 20
3.2.4 Collaborative work . . . . . . . . . . . . . . . . . . . . . 20
3.3 Construct3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.2 Basic object types . . . . . . . . . . . . . . . . . . . . . 22
3.3.3 Geometric operations . . . . . . . . . . . . . . . . . . . . 23
3.3.4 New functions . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 ACIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4.1 Boundary representations . . . . . . . . . . . . . . . . . 24
3.4.2 Geometric representations . . . . . . . . . . . . . . . . . 26
4 Geometric Background 27
4.1 Numerical Derivation Methods . . . . . . . . . . . . . . . . . . . 27
4.1.1 Dierentiation . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.2 Higher derivatives . . . . . . . . . . . . . . . . . . . . . . 28
4.1.3 Taylor expansion . . . . . . . . . . . . . . . . . . . . . . 28
4.1.4 Numerical approximation by dierence quotients . . . . . 29
4.1.5 Advanced methods of numerical approximation . . . . . 31
4.2 Curve representation . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.1 2D curve representation . . . . . . . . . . . . . . . . . . 32
4.2.2 3D curve representation . . . . . . . . . . . . . . . . . . 35
4.3 Dierential geometry on a curve . . . . . . . . . . . . . . . . . . 36
4.3.1 Frenet frame in curve points . . . . . . . . . . . . . . . . 36
4.3.2 Center, circle and sphere of curvature . . . . . . . . . . . 40
4.3.3 Plane of curvature . . . . . . . . . . . . . . . . . . . . . 43
5
Contents
4.4 Dierential geometry on a surface . . . . . . . . . . . . . . . . . 45
4.4.1 Surface curvature . . . . . . . . . . . . . . . . . . . . . . 45
4.4.2 Types of surface curvature and special points . . . . . . 47
4.4.3 Frenet frame in surface points . . . . . . . . . . . . . . . 47
4.4.4 Meusnier point . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 Geometric properties of implemented surface classes . . . . . . . 49
4.5.1 Helical transformations . . . . . . . . . . . . . . . . . . . 49
4.5.2 General sweeps . . . . . . . . . . . . . . . . . . . . . . . 53
5 Design 54
5.1 Construct3D startup . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Workow for creating a new object . . . . . . . . . . . . . . . . 54
5.3 Dierential geometry functions . . . . . . . . . . . . . . . . . . . 55
5.3.1 Frenet frame . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3.2 Plane of curvature . . . . . . . . . . . . . . . . . . . . . 56
5.3.3 Center and circle of curvature . . . . . . . . . . . . . . . 57
5.3.4 Meusnier point . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Sweep functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4.1 Helical sweeps . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4.2 General sweeps . . . . . . . . . . . . . . . . . . . . . . . 61
5.5 The Personal Interaction Panel (PIP) . . . . . . . . . . . . . . . 61
5.5.1 Extension of the user interface . . . . . . . . . . . . . . . 61
5.5.2 Keyboard shortcuts for the desktop setup . . . . . . . . 63
6 Implementation 64
6.1 Class overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2 Object creation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Dierential geometry functions . . . . . . . . . . . . . . . . . . . 66
6.3.1 Constructor and destructor . . . . . . . . . . . . . . . . 66
6.3.2 Common precalculations . . . . . . . . . . . . . . . . . . 66
6.3.3 SoPointKit . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3.4 SoCurveKit . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3.5 SoPlaneKit . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.4 Sweep functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.4.1 SoCurveKit . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.4.2 SoSurfaceKit . . . . . . . . . . . . . . . . . . . . . . . . 70
6.5 Undo/redo list . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.6 Extension of the user interface . . . . . . . . . . . . . . . . . . . 71
6.6.1 Routing of user interface events . . . . . . . . . . . . . . 74
7 Results 76
7.1 Dierential geometry functions . . . . . . . . . . . . . . . . . . . 76
7.1.1 Frenet frame . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.1.2 Plane of curvature . . . . . . . . . . . . . . . . . . . . . 77
6
Contents
7.1.3 Center and circle of curvature . . . . . . . . . . . . . . . 78
7.1.4 Meusnier point . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Sweep functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.2.1 Helical sweeps . . . . . . . . . . . . . . . . . . . . . . . . 80
7.2.2 General sweeps . . . . . . . . . . . . . . . . . . . . . . . 82
8 Conclusion 83
List of References 84
7
1 Motivation
1 Motivation
Mathematics and descriptive geometry are subjects in high school which will
either be loved or hated. On the one hand there are the students who have
the luck to really understand most of what they have to do in these subjects
but for most of the students mathematics and geometry are closed books.
Fortunately I was one of the former and so in the 11th grade, I decided to
take descriptive geometry instead of an emphasis in natural sciences. Rapidly
it became on of my favorite subjects. Because of my ability to visually imagine
my tasks I easily understood the subject.
Construct3D, an augmented reality application for mathematics and geom-
etry education, is a tool which was mainly developed to improve the spatial
abilities of its future users, high school and university students. The user has
the opportunity to actually see in 3D what he only has to imagine in conven-
tional geometry classes. The traditional pen and paper based methods used
for construction and even computer based visualizations only provide 2D pro-
jections of 3D content. The advantage of computer based approaches over
pen and paper based methods is that the students can really concentrate on
the construction and geometric understanding rather than to achieve a perfect
drawing.
In Construct3D users achieve a real 3D impression of the virtual geomet-
ric content projected in a real environment. Users can explore the space of
modeling by walking around the constructions and looking at them from dif-
ferent points of view. Construct3D is also a dynamic tool so the user has the
possibility to change a part of the construction and in real time the whole
constructions also changes. The user gains a deeper insight in the relation-
ship between the parts of the construction. In our geometry room in high
school there was a cone for demonstrating the conic sections. It was built
out of plastic in dierent colors and could be disassembled. For the purpose
of demonstrating the conic sections it was sucient and useful but there are
other geometric constructions which are more dicult to understand. Some-
times the possibility to dynamically change parameters of the constructions
provide a better understanding of the subject.
For understandable reasons choosing a eld concerned with geometry edu-
cation was selfevident for me being a really useful combination of informatics
and geometry. Implementing dierential geometry functions was interesting
because these functions wereup to nowrarely implemented in conventional
geometry packages and if, they are not really dynamic. Dierential geometry
in a dynamic way in an augmented reality application was completely new.
I hope with my implementation of generalized sweep objects and functions of
dierential geometry I will help future users of Construct3D to gain a deeper
understanding of these elds and to have more fun with geometry.
8
2 Related Work
2 Related Work
This section should give an overview over some software packages which are
related to the work in this thesis. There will be a focus on dynamic geometry
software but there are also connections to other software for dierential geom-
etry or CAD applications. In this work rst software packages for dynamic
geometry for working in 2D and mostly for educational purposes will be in-
troduced. After that follow software packages for three dimensional geometry.
Finally some CAD packages which provide dierential geometry functions will
be presented.
2.1 History of dynamic geometry software
For a very long time geometric constructions were created only with the aid of
pencil and ruler. Static visualizations were generated and later changes in the
construction were not possible. The instructions at school were augmented
only with a few models with exible parts. These models were expensive
and often only the teacher was allowed to work with them. Therefore, the
students had mostly no possibility to examine and explore the models. In
the seventies of the previous century lm material was sometimes used to
demonstrate dynamic behaviour. But their development was complex and
expensive and the students were again only observers.
With the geometry software of the rst generation in the eighties one could
do exactly the same things as one could do with pen and paper. Therefore, an
added value could not be identied and this type of software was rarely used.
At the beginning of the nineties the rst dynamic geometry software pack-
ages were developed, e.g. Cabri Géomètre and Geometer's Sketchpad. The
characteristic feature was the so-called "drag mode". The base points of a ge-
ometric construction could be dragged only with the help of the mouse without
the need of keyboard input. This was and still is an intuitive way to change
the parameters of a geometric construction dynamically. The interrelation-
ship between the construction's components will not be aected through the
change of the base points and at any time the user has a mathematical valid
construction. At that point of the development the construction task itself lost
its importance and the exploration and examination became the work's main
part. The students got the chance to explore the concepts themselves, every-
body in its own speed and way. The student's role changed from the passive
observer to that of the active user and explorer of geometry. The construction
process itself became a lot easier and the remaining tasks were to understand
the relationships and to draw the correct conclusions. Another main charac-
teristic of dynamic geometry software is the possibility to generate so-called
"loci". Loci originate from the trace of a construction point B when another
point A of the construction is moved and aects point B. In the course of time
9
2 Related Work
also the generation of interactive worksheets for students became more impor-
tant. Instead of starting with a blank worksheet the students get a specic
task to solve and can check their results themselves. Often the installation
of the software is not needed anymore because most software packages run as
applets in a web browser.
Initially dynamic geometry software was developed only for geometry in
two dimensions. But at the beginning of the 21st century dynamic geometry
software was also developed for 3D. Features and handling are very similar but
sometimes more dicult and complex because the mouse as 2D input device
has to deal with three spatial directions.
Construct3D as an augmented reality application takes this progress a step
further and leads the geometry students to a real 3D environment where they
can explore geometry problems in their genuine space. A main goal of the use
of dynamic 3D geometry software is to improve the student's spatial abilities.
At the moment, there exist several research projects for analyzing the concrete
eects of dynamic 3D geometry software on spatial abilities but rst tests and
projects have already shown that students really benet. [6, 7]
2.2 Dynamic geometry software
In this section several software packages will be introduced. Neither oers
all the important features of Construct3Dthree dimensional construction,
examination in augmented reality, interactive behavior, changes applying in
real time and support of dierential geometrybut inspired in some way the
concept of Construct3D and its features. Each of these software packages
has its own feature set and strengths and could be used for partially solving
the problems concerning dierential geometry which can both be solved and
properly visualized to full extent only with Construct3D by now.
2.2.1 Dynamic 2D Geometry software
Euklid DynaGeo and Cinderella are popular and widely used examples of dy-
namic 2D geometry software. These two software packages have several fea-
tures in common mainly that they are interactive and changes are applied in
real time. A further relatively new dynamic 2D geometry software is GeoNext
[31], developed by the University of Bayreuth, Germany, since 2003. The
software package GeoGebra [13] is a so far unique combination of a dynamic
geometry software and an algebra software.
Euklid DynaGeo Euklid DynaGeo [20] is a shareware software for dynamic
2D geometry for Windows (see gure 1) developed by Roland Mechling since
1994.
10
2 Related Work
Figure 1: This gure shows a screenshot of Euklid DynaGeo [20]. Below the
menu bar there is a tabbed bar with buttons for the main functions,
construction, form and color, measurement and calculation. The
screenshot shows the construction of a circle's circumcircleMu. The
input elements, the straight lines a, b and c between the points A,
B and C are colored in green, red and blue.
This is a popular software for geometry, mathematics and descriptive geometry
education, mainly used in the lower grades of high schools. It was one of the
rst available software packages aiming with its high usability especially at
beginners with geometry programs.
As in every dynamic geometry software in Euklid DynaGeo some base points
of the constructions can be dragged with the mouse without loosing the in-
terrelationship between the components of the construction. Constructing a
triangle's circumcircle it will always pass through all corners of the triangle,
regardless where a corner of the triangle is moved.
The generation of dynamic loci gives answers to questions like "how does a
point on a bicycle tire move in space when the bike is in motion?" Macros which
combine multiple construction steps can be generated in an easy point-and-
click manner. All lines and points can be colored and labeled freely. Distances
and angles between the objects can be measured and construction dependent
terms can be calculated. Animations can be generated and the projects can
be exported to various le formats.
11
2 Related Work
Cinderella The Cinderella project [16] is the product of a sequel of three
projects done between 1993 and 1998. Cinderella is a dynamic 2D geometry
software developed by Ulrich Kortenkamp and Jürgen Richter-Gebert since
1998. It is written in Java and therefore available for Windows, Linux and
Mac.
Figure 2: This gure shows an interactive student exercise in Cinderella [25].
The input elements are marked in red. At the bottom there are tools
that can be used to solve the exercise "construct the circumcenter of
a triangle using only straightedge and compass". There are also but-
tons for giving hints and for testing the correctness of the student's
solution.
In its so-called "drag mode" the parts of a construction can also be moved
with the mouse without destroying the interrelations between the components
of the construction. No keyboard input is necessary. In its main features
Cinderella is very similar to Euklid DynaGeo but provides several additional
features. Unlike Euklid DynaGeo Cinderella not only supports Euclidean ge-
ometry but also non-Euclidean geometry, like hyperbolic and elliptic geometry.
The problem of jumping elements resulting from ambiguities (e.g. a circle
and a line can have none, one or two intersections) was solved in implementing
the Cinderella kernel in a complex vector space. To avoid singular situations
results of analytic function theory were used. This system established the
base of a reliable randomized theorem checking, which tests automatically
the correctness of geometric theorems. It became also possible to generate
complete geometric loci.
In its actual version 2.0 Cinderella was augmented with several new features
which were also motivated by user's suggestions. Cinderella now also supports
12
2 Related Work
some kind of macros. A newly added feature are transformations and transfor-
mation groups which shall support the user at more advanced constructions. A
quite independent part is the so-called CindyLab, a physics simulation engine
which provides a particle, mass and force simulation paradigm. New is also
CindyScript, a full-featured, mathematically oriented, high level programming
language. An interesting new feature is also the support for the recognition
of hand sketches, so Cinderella could be used with pen and tablet, an inter-
active whiteboard or a PDA. It is also possible to create interactive exercises
for students while providing only selected functions that students shall use for
solving a problem (see gure 2) [16, 17, 25].
2.2.2 Dynamic 3D Geometry software
Archimedes Geo3D and Cabri 3D are examples for dynamic 3D geometry soft-
ware. They extend the concept of the software of the previous section to three
dimensional space. Meanwhile they share the same basic concepts of real time
execution and interaction.
Archimedes Geo3D Archimedes Geo3D [11] has been developed by Andreas
Goebel since 2006 and is available for Windows, Linux and Mac (see gure 3).
It expands the idea of dynamic geometry to the third spatial dimension. Its
idea of moving the base points of a construction, the "drag mode", is the same
as for dynamic 2D geometry programs. At the beginning, the handling of a
dynamic 3D geometry software is more complicated because the mouse as a 2D
input device has to be combined with pushing the keyboard buttons <shift>
and <strg> to navigate in three spatial dimensions. Archimedes Geo3D sup-
ports also an extended drag mode where all (not only the base objects) can
be moved and also rotated freely. The handling of the program is simplied
through the possiblity of using keyboard shortcuts.
Like the 2D programs Archimedes Geo3D supports the creation of loci, the
traces of points, i.e. curves. Because of the third spatial dimension it is also
possible to create traces of curves, i.e. surfaces. Input cannot only be given
through the creation of points and basic shapes but also through entering of
mathematical terms. Therefore Archimedes Geo3D supports also vector anal-
ysis as well as curves and surfaces can be dened through their equations.
Further available features are texturing, animation creation and shadow gen-
eration. Macros can be used to record multiple construction steps and can also
be called recursively. Finally, Archimedes Geo3D supports "real" 3D display
with the aid of anaglyph images or shutter glasses. [12]
Cabri 3D Cabri 3D [3] is a dynamic 3D geometry software developed by
Cabrilog. The rst Cabri software programs were developed in the research
13
2 Related Work
Figure 3: This gure shows a screenshot of Archimedes Geo3D [11]. At the
top the menu bar is located with options for le actions, object con-
struction, measurement, calculations and macro generation. Below
the toolbar can be seen which shows buttons for actions available
for the current selection. On the left there is an object box where
objects can be selected and deselected. With a double click on a
point its coordinates can be changed.
labs of the France Centre National de la Recherche Scientique (CNRS) and
at the Joseph Fourier University in Grenoble. In 1985 Jean-Marie Laborde,
co-founder of Cabrilog, had the vision to make 2D geometry easier to learn for
students and for teachers more enjoyable to teach.
A Cabri 3D document consists of a set of pages with one or more views
on each page that can be freely manipulated. The user can choose from over
fteen standard projections and can change the viewpoint on a construction
unrestrictedly. The pages can be enriched with comments in rich text and the
constructions can be augmented with colors, textures and graphic styles (see
gure 4). Lenghts, angles, areas and volumes can be measured and further
calculations can be performed with these results. Expressions can be created
using algebraic concepts like numbers, variables and operations. Animations
can be used for modeling physical phenomena. A tool replays the user's pre-
viously performed construction steps. A nice feature is the unfolding of all
polyhedra into a printable net.
The le format of Cabri 3D is based on the XML standard (CabriML) and
14
2 Related Work
Figure 4: This gure shows a screenshot of a Cabri 3D construction. It demon-
strates that the apexes of three double cones (yellow) which are dis-
tinct and tangent to two among three spheres (purple) are collinear.
is therefore easy to understand and to modify. High resolution images can
be copied to the clipboard. Every project can be exported as interactively
manipulable gure to a web page. The required plugin is available for free for
Windows. These gures can also be incorporated in Microsoft Oce docu-
ments. [3, 27]
2.2.3 CAD software providing dierential geometry functionality
Rhinoceros and Pro/Engineer are examples of commercial CAD software that
also supports dierential geometry and are therefore related to this work. Both
software packages have in common that they are not interactive in a sense
that changes are not applied in real time compared to the software packages
presented in the previous two sections.
Rhinoceros Rhinoceros [19], abbreviation Rhino (see gure 5), is a com-
mercial software for NURBS (non-uniform rational B-Splines) modeling for
Windows developed by McNeel. Rhinoceros is used by constructors, architects
and designers. A particularity of Rhino is that all surfaces are constructed
of NURBS. With Rhinoceros NURBS curves, surfaces or volumes can be cre-
ated, edited, analyzed or converted, regardless of their complexity, grade or
size. Polygon meshes and point clouds are also supported.
15
2 Related Work
Figure 5: This gure shows a screenshot of the Rhino 3D [9] user interface.
The shown object can be analyzed with a wide range of analysis
tools for surface curvature and geometric continuity.
Rhino supports 3D freeform modeling without any restrictions and is never-
theless extremly precise. Rhino is also compatible with a wide range of other
CAD or modeling software packages and can therefore be seen as a connector
between other programs. It supports also a variety of input and output devices
like 3D digitizers, 3D scanners and 3D printers.
The link to this thesis is Rhino's support of dierential geometry functions.
There are analysis tools for surface curvature, geometric continuity, deviation,
curvature graphs on curves and surfaces, Gaussian curvature, mean curvature
and minimum or maximum radius of curvature.
Helices and spiral curves can be constructed. Rotational surfaces and extru-
sions are also available. It also supports some forms of curvature measurement.
In its latest version 4.0 many features of Rhino were enhanced, including
texturing, rendering and animation. Advanced custom display modes are now
supported, as well as dual-screen support and stereo display.
Pro/Engineer Pro/Engineer [21] is a professional parametric 3D CAD soft-
ware package also known as ProE or Pro/E developed by the Parametric Tech-
nology Corporation (PTC). It is available for Windows, Linux, Irix, HP-Unix
16
2 Related Work
Figure 6: Screenshot Pro/Engineer
and Solaris.
In Pro/Engineer all objects are constructed in 3D and in succession the
corresponding drawings can be extracted. The term parametric means that
everything has a dimension and a change in a part of model changes the whole
model geometry. It is also bidirectional associative, so every change in the
model geometry follows a change in all abbreviations of the model like drawings
and constructions groups and vice versa.
The main application areas are the automobile industry (construction of
motors and gearboxes) and machine construction (see gure 6).
17
3 Technological Foundations
3 Technological Foundations
This sections shall explain the components which build the basement of Con-
struct3D and therefore of the new implemented functions.
3.1 Open Inventor and Coin3D
The Open Inventor rendering library is a framework of C++ classes that im-
plement a scene graph based rendering library using OpenGL. It supports an
event driven programming style where the application is normally composed
as a set of call back functions that react to framework events. [23]
It is a library of objects and methods for creating interactive 3D graphics
applications. So it oers a range of objects which can be used, modied and
extended. There are three types of objects: database primitives, interactive
manipulators and components.
All information of the created 3D objects e.g.shape, size, color, texture,
locationis stored in a scene database. This approach oers the exibility not
only to render these objects on a screen but also to move the objects, view
them from dierent viewpoints, change their properties, to highlight, animate
and to interact with them. The Open Inventor programming model is intuitive
to use because it is based on the "real world" we live in. [32, 33]
Open Inventor originally was developed by SGI but after it's release under
Open Source license in 2000 SGI discontinued the further development. In
1995 the company Systems In Motion started the development of Coin as a
3D graphics rendering library. After a few years Coin was rewritten from
scratch and "inventorized" and now implements the whole Open Inventor API
including a number of extensions. For Free Software development Coin3D is
available under the GNU GPL license. [30]
3.1.1 The Open Inventor Toolkit
At the programming level Open Inventor oers the following tools:
 3D scene database: used to create a hierarchical 3D scene. (shape, prop-
erty, group, engine and sensor objects)
 Node kits : prebuilt groups of Inventor nodes.
 Manipulators : objects in the scene database users are able to interact
with. (handle box, trackball)
 Inventor component library : library for high-level interactive tasks (ren-
der area, material editor, viewers, utility functions) [32]
18
3 Technological Foundations
3.1.2 The Scene Database
The scene database consists of basic elements called nodes. A node is a C++
class type with additional functions that aggregates objects called elds that
store a value of a certain type (e.g. string or integer). A node contains infor-
mation like surface materials, shape descriptions or geometric transformations.
Every 3D shape or light source itself is also a node. To form a hierarchical
structure a node of type SoGroup can associate a list of other nodes, called
children. The children of a group node are ordered and can be accessed from
left to right with index 0 up to n−1. The ordered collection of all these nodes
is the scene graph which is stored in the database. A mechanism called action
traverses the scene graph recursively to compute data.
Classes of database primitives include the following:
 Shape nodes : sphere, cube, cylinder, cone, quad mesh, etc.
 Property nodes : material, lighting model, textures, etc.
 Group nodes : separator, switch, etc.
 Engines : Engines can be connected to other objects to animate parts of
the scene or otherwise constrain some parts of the scene.
 Sensors : A sensor can detect changes in the underlying database or
reacts to a timer and in succession calls a function. [23, 32]
3.1.3 Node Kits
A node kit is a collection of nodes with a specied arrangement which helps
to keep things in order. There are some sort of templates telling you which
kinds of nodes can be added to a node kit and where to place them. There is
also the possibility to extend Open Inventor in creating new customized node
kits. [32]
3.2 Studierstube
Studierstube [26] is a research software system for augmented reality applica-
tions. It consists of a set of extension nodes to the Open Inventor [32] (see
section 3.1) rendering library. It includes support for interaction based on 3D
tracking events. There are rendering and output modes for virtual and aug-
mented reality output devices. It also provides tools for developing distributed
applications and user management functions for multiple user support.
3.2.1 3D event system
Studierstube extends the Open Inventor library to support not only standard
desktop input devices such as keyboard and mouse but also trackers with six
19
3 Technological Foundations
degrees of freedom (position and orientation in the space). These user interface
events are handled with a dedicated Handle3DEventAction. The base class
Base3D implements the basic methods called during the traversal of the scene
graph after a user interface event.
3.2.2 Widget system
Widgets are graphical objects that react to input events and change their state.
A state change means a dierent graphical representation (e.g. a released
button is a box with a given height and a pushed button is a box with a
lower height and a dierent color) and a change to the elds of the widget
(e.g. a Boolean is set from false to true). Studierstube implements a standard
set of widgets such as toggle, push and radio buttons, lists and sliders which
are represented by 3D geometry. The personal interaction panel (PIP) is a
tracked tablet overlaid with a traditional 2D graphical user interface consisting
of widgets. It combines the advantages of a physical representation, the natural
way of interaction with the virtual widgets and the haptic feedback when an
interaction device, mainly a pen, collides with the tablet with the exibility of
switching between dierent sets of widget groups.
3.2.3 Dynamic application loading
Applications are implemented as a sub scene graph in a Studierstube process
and can be dened by implementing a new application node class. On the one
hand the application can use any existing sub scene graph to create required
elements, on the other hand own specialized node types can be dened. All
the time all Open Inventor operations can be used on the application data.
Because of Open Inventor's support of serialization of a scene graph to and
from a le, an application can be loaded and saved at runtime. At any time
the application's scene graph represents the current state of the application
because all data is stored in elds or nodes of a sub scene graph.
3.2.4 Collaborative work
Studierstube provides the necessary functionality for collaborative work. It
supports the simultaneous connection of several display devices which can
provide personalized views and also supports an unlimited number of input
devices. A typical dual user setup consists of two head mounted displays
(HMD), two interactions devices (pen) and two personal interaction panels
(PIP), that is six tracked devices and two output windows. Because of the
tracking of the input devices the user can see the scene through his HMD from
his own viewpoint. It's also possible to render a private sub scene graph for
each user which gives the possibility of a personalized view. [23]
20
3 Technological Foundations
3.3 Construct3D
Construct3D [14] is a 3D dynamic geometry construction tool. It is based on
the Augmented Reality System Studierstube. Users see the real environment
augmented with virtual content. That gives the users, mainly high school and
university students, the opportunity to actually construct, explore and inter-
act with three dimensional content in "real" 3D space (see gure 7). Until
now students had to construct geometry with traditional pen and paper based
methods or with desktop computer software that also only shows a 2D pro-
jection of a 3D content. Construct3D oers the opportunity to walk around
a geometric construction and view it from dierent angles. It is also a col-
loborative Augmented Reality application and can be used by several users
simultaneously which can interact with each other.
Figure 7: This gure shows two users of Construct3D working together on
a three dimensional construction. The users wear head mounted
displays (HMD) and interact with the construction with the aid of
pen-based input devices. A tracked physical tablet with a projection
of a menu system, the Personal interaction panel (PIP), is used to
access Construct3D's functions.
A further key feature of Construct3D is the support for dynamic 3D geome-
try. By moving one part of the construction the other parts of the construction
adjust themselves in real-time. The relationships between the components of
the construction can be explored and a deeper understanding of the matter
may be achieved. Construct3D is also a tool that should improve the spatial
abilities of the users [6].
21
3 Technological Foundations
3.3.1 Design
At the moment Construct3D oers functions for the construction of points and
two and three dimensional objects and also for planar and spatial operations
on these objects. It also provides functions for measuring, structuring content
into layers and basic system functions.
At the start up a 3D window with innite size is initialized to cover all
available space. The users work with a user interface consisting of the Personal
Interaction Panel (PIP) and a pen-based input device. The PIP is a tracked
physical tablet with a projection of a menu system on it. The pen is used for
manipulating the PIP but also for direct manipulation of the objects.
In point mode the user points with the pen at the location where he wants
a point to be and clicks and at an absolute position in space a point will be
created. When point mode is turned o the nearest point or object in the scene
will be highlighted. Then the user can select points or objects with the pen and
can create new objects with the previously selected objects as input elements.
After selecting an object the user can interact with them, e.g. selecting and
moving a point on a circle and the circle will automatically adjust its size
and position. A preview function available for most constructions shows the
resulting object in wireframe mode. The 3D constructions can be projected on
orthogonal planes to provide classical 2D views of the 3D content (e.g. ground
view, upright projection, prole).
When the application is used with several users, each user's constructed
points and objects get an unique color. Nevertheless each user can modify
constructions of another user.
Every construction step is stored in a command list and an undo/redo history
oers the possibility to move backward and forward in the construction history.
3.3.2 Basic object types
Construct3D oers a range of basic object types:
 Points: freely positioned or xed on curves or surfaces
 Lines
 Planes
 Circles and ellipses
 Cuboids
 Spheres
 Cylinders
 Cones
22
3 Technological Foundations
 B-Spline curves with control points and interpolated B-Spline curves
 NURBS surfaces with control points and interpolated NURBS surfaces
 Sweep surfaces
3.3.3 Geometric operations
The following geometric operations can be performed on the basic object types
and also on more complex objects generated throughout the construction pro-
cess:
 Boolean operations (union, dierence, intersection)
 Intersections
 Planar slicing of objects
 Rotational sweep around an axis
 Surface normals
 Tangential planes
 Tangents
 Common tangent to two circles
 Plane normal to a line
 Line normal to a plane
 Plane of symmetry
 Angle bisector
 Mid point [14]
3.3.4 New functions
As result of this diploma project the following new functions have been imple-
mented which will be explained later in the sections 4.3, 4.4 and 4.5:
Dierential geometry
 Center of curvature
 Circle of curvature
 Plane of curvature
 Meusnier point
 Frenet Frame
23
3 Technological Foundations
Sweeps
 Helical sweeps
 General sweeps
3.4 ACIS
Geometric modeling is a very dicult task concerning higher mathematics and
complex programming issues and also practical problems like numerical accu-
racy. Therefore after considering to implement these functions themselves the
developers of Construct3D decided to use the 3D geometric modeler ACIS [28]
which is widely used from CAD software to virtual reality applications. The
ACIS modeler is a software library developed by the Spatial corporation for
representing and manipulating shapes. It features an open, object-oriented
C++ architecture that enables robust 3D modeling capabilities. ACIS inte-
grates wire frame, surface and solid modeling functionality with both manifold
and non-manifold topology and geometric operations. [4]
3.4.1 Boundary representations
ACIS is an exact geometric modeler that represents shapes by modeling their
boundaries. This representation is called boundary representation or B-rep for
short because ACIS calculates the equations of the curves and surfaces that
lie on the boundary between the inside and the outside of a solid 3D object.
Imagine a thin wooden plate with a circular hole in it. This board consists
of the following surfaces: two planar surfaces on the upper and the lower side
of the board, four planar surfaces on the exterior and a cylindrical surface
for the hole. The intersections of these surfaces form curves and the meeting
points of the curves form points. So a boundary representation consists of
vertices (points), edges (curves), faces (surfaces). The only problem with this
approach is that, with a few exceptions e.g. points, circles and spheres, classical
analytical geometry has no boundaries. The solution for this problem is to
explicitly dene these boundaries.
Curves are bounded by pairs of points lying on the curve and surfaces are
bounded by curves lying on the surface. The relationships of the geometric
elements are called topology and can be visualized as a graph where the nodes
represent points, curves and surfaces. Also the sharing of bounding entities
(e.g. the corners of a cube bound three dierent edges and each edge bounds
two faces) can be stored in such a graph. These entities are called topological
entities because they dene how things connect:
 The shape of a face is a surface bounded by edges associated with it.
24
3 Technological Foundations
 The shape of an edge is a curve bounded by a pair of vertices associated
with it.
 The location of a vertex is the position of a point.
Only the three topological entities face, edge and vertex are really needed
but in practice the introduction of a few more topological entities is useful.
Below there is an explanation how a path through the graph shown in gure
8 can be read:
 Coedge: In most cases edges lie between two faces but there are opera-
tions which need only the boundary of an individual face. So an edge is
associated with a number of coedges, one coedge for each face.
 Loop: The connection of coedges which form a high-level representation
of a faces boundary is called loop.
 Face: A collection of loops can be summarized to a face.
 Shell : The connection of faces which form a high-level representation of
a surface is called shell.
 Lump: A collection of shells which dene a separate piece of an object
is called lump.
 Body : A collection of lumps can be summarized to a body.
Finally there are a few restrictions to the ACIS boundary representations to
guarantee only manifold objects:
 Every edge must lie between two faces.
 Faces and edges do not self intersect.
 Every entity in the model is bounded. [4]
Figure 8: Boundary representation hierarchy in ACIS. A possible path through
this graph (bold line) is described in section 3.4.1.
25
3 Technological Foundations
3.4.2 Geometric representations
There are several ways to represent geometry, e.g.:
1. Explicit: y = f(x)
2. Implicit: f(x, y) = 0
3. Parametric: x = f(t), y = f(t)
The explicit form is the best known and least useful representation because of
its numerous possible special cases. For geometric modeling the implicit form
which denes two half-spaces is more useful. Half-spaces can be imagined to
be generated when an innite plane cuts the 3D space in two. Given a point's
coordinates on the boundary, the implicit equation evaluates to zero. On either
side the equation evaluates to more or less than zero. The drawback of the
implicit representation is that it cannot be easily derived for all surfaces used
by ACIS.
The third form, the parametric representation, can be created for all surfaces
used by ACIS. The parametric form of a curve is dened by equations given
in terms of a single independent variable t. A mathematical function with two
parameters u and v can be used to dene every point on a surface. That means
every surface has its own (u, v) coordinate system.
Because of its specic advantages and drawbacks ACIS uses analytic, implicit
and parametric geometry representations. The analytical curves and surfaces
have a low level implicit representation as well as higher level parametric repre-
sentation. General smooth curves and surfaces, called splines, are represented
through parametric representation. There are several forms of splines but all
are constructed by specifying a number of control points which determine their
shape. [4]
26
4 Geometric Background
4 Geometric Background
Elementary dierential geometry deals with curves and surfaces in three di-
mensional space and their curvature properties. Dierential geometry can
be seen as a synthesis between analysis, in particular innitesimal calculus,
and elementary geometry. When handling dierential geometry problems nu-
merically, the need arises to calculate derivatives of complicated functions.
Therefore knowledge on numerical derivation methods is useful. For a com-
prehension of the concepts of dierential geometry a basic understanding of
these topics is required.
In the following section the mathematical and geometrical foundations are
explained which are needed for understanding the dierential geometry tools
implemented for this thesis. At the beginning a short overview over dier-
ent numerical derivation methods is given which are fundamental because all
dierential geometry tools need at least the second derivation of a curve's
function. After this the parametric form of a curve representation both in two
and in three dimensions will be introduced. Next, the dierential geometry
concepts on a curve are explained: the Frenet frame and the center, the circle,
the sphere and the plane of curvature. Afterwards the concepts for surfaces
are introduced, starting with an explanation of surface curvature and their
types, followed by the Frenet frame in surface points and the Meusnier point.
Finally the geometric properties of the implemented surface classeshelical
sweeps and general sweepsare introduced.
Detailed descriptions of the mathematical and geometrical foundations of
this thesis can be found in [15, 29, 34].
4.1 Numerical Derivation Methods
In this section some concepts of calculating derivations of a function in a
numerical way will be presented. Unfortunately it is unknown how ACIS
performs the derivation of functions and therefore it is also unknown how
Construct3D performs these calculations. The numerical derivation methods
presented in the following shall only give a short overview to get an idea how
ACIS could perform these tasks.
4.1.1 Dierentiation
The derivation of a function f(x) at a point x is the linear image of the
function which approximates the change of the function best. The geometrical
correspondent of the rst derivative of a function at a given point is the gradient
of its tangent in that point. To estimate the gradient of the tangent we can
27
4 Geometric Background
assume the gradient of a secant, a straight line through two points of the
function curve
f ′(x) ≈ f(x+ h)− f(x)
h
(1)
The gradient of the secant is called dierence quotient. Then we let the
distance between the points f(x) and f(x+ h) converge to zero. A function is
called dierentiable at position x if the limit
lim
x→x0
f(x)− f(x0)
x− x0
= lim
h→0
f(x0 + h)− f(x0)
h
(2)
exists. This limit is called dierential quotient or derivative of f with respect
to x and can also be written as f ′(x0) (Lagrange's notation) or
df
dx
(x0) (Leibniz's
notation).
The derivatives of all elementary functions can be calculated considering the
concepts of the dierence and the dierential quotient but in daily practice
the derivatives and the antiderivatives are well known or can be looked up in
a reference book. [1, 22]
4.1.2 Higher derivatives
If the derivative of a function f is derivable, a second derivative can be obtained
as derivative of the rst derivative. A function f can be derivable up to n times,
i.e. up to its n-th derivative. [1]
f ′′ = f (2) =
d2f
dx2
, f ′′′ = f (3) =
d3f
dx3
, . . . , f (n) =
dnf
dxn
(3)
4.1.3 Taylor expansion
The so-called Taylor expansion is used to approximate complicated functions
by a series of polynoms. Often a function can be fairly good approximated by
a Taylor expansion truncated after only a few terms. Given a (n + 1) times
continuous derivable function f in its interval I then we can propose for all a
and x within I the Taylor formula
f(x) = Tn(x) +Rn+1(x) (4)
where
28
4 Geometric Background
Tn(x) =
nX
k=0
 
f (k)(a)
k!
(x− a)k
!
=f(a) +
f ′(a)
1!
(x− a) +
f ′′(a)
2!
(x− a)2 + · · ·+ f (n)(a)
n!
(x− a)n
(5)
with the remainder term
Rn+1(x) =
1
n!
Z x
a
(x− t)nf (n+1)(t)dt (6)
A function f that is indenitely derivable is called a smooth function and
the Taylor formula can be expanded to the Taylor series. The Taylor series is
a power series of a real or complex function f that is innitely derivable in a
neighborhood of a real or complex number a [1]
f(a)+
f ′(a)
1!
(x−a)+
f ′′(a)
2!
(x−a)2 + · · ·+ f (n)(a)
n!
(x−a)n =
∞X
n=0
f (n)(a)
n!
(x−a)n
(7)
4.1.4 Numerical approximation by dierence quotients
A numerical solution is necessary if either a precise solution is not available or
an exact solution can not be computed because of its huge costs. There are
some methods of calculating a derivative numerically and for each problem one
has to balance the pros and cons concerning cost and accuracy of the solution.
When computing the derivative of a function there are several potential sources
of error, e.g. the round o error and the truncation error.
Error estimate for a asymmetrical dierence quotient Assuming one
chooses an arbitrary small number for h, e.g. h = 0, 0001 and use the equations
of section 4.1.1 h has no exact binary representation, neither has x+h in most
cases. Therefore these values are represented with some fractional error, called
round o error, depending on the oating point precision of the machine where
the calculation is executed. As a consequence there is also a fractional error
in the derivative and in the following calculations. To avoid this error one has
to choose a value h where it is guaranteed that h can be exactly represented.
If h is exactly represented the round o error of the dierence quotient is
er ∼ ef
f(x)
h
 (8)
29
4 Geometric Background
where ef is the fractional accuracy with which f is computed. For a simple
function the fractional accuracy is comparable to the machine accuracy ef ≈
em.
The truncation error comes from higher terms in the Taylor expansion. For
the function f(x+ h) one can assume the following Taylor expansion
f(x+ h) = f(x) + hf ′(x) +
1
2
h2f ′′(x) +
1
6
h3f ′′′(x) + . . . (9)
Therefore one can assume a Taylor expansion for the dierence quotient:
f(x+ h)− f(x)
h
= f ′(x) +
1
2
hf ′′(x) + . . . (10)
The truncation error of this Taylor expansion is in the order of
et ∼ |hf ′′(x)| (11)
If h is varied to minimize the sum of errors er +et one can assume the optimal
value of h to be
h ∼
s
eff
f ′′
≈ √efxc (12)
where
xc ≡
s
f
f ′′
(13)
is the curvature scale or characteristic scale of the change of function f . If
no further information is available a rst assumption is xc = x. If x is nearly
zero a dierent assumption should be used. With the proposition h ≈ √efxc
the fractional accuracy of the computed derivative can be estimated
(er + et)
|f ′|
∼ √ef
s
ff ′′
f ′2
∼ √ef (14)
From this it follows that the dierence quotient from section 4.1.1 gives at
best the square root of the machine accuracy dependent on the machine's
oating point precision. [22]
30
4 Geometric Background
Error estimate for a symmetrical dierence quotient A more accurate
result can be achieved if it can be aorded to calculate two functions instead
of one. Then the symmetrical form of the dierence quotient can be used
f ′(x) ≈ f(x+ h)− f(x− h)
2h
(15)
The truncation error for this form of dierence quotient is et ∼ h2f ′′′ and
the round o error stays the same as before. In this case the optimal choice of
h is
h ∼ 3
s
eff
f ′′′
∼ 3
√
efxc (16)
Therefore the fractional error ef is
er + et
|f ′|
∼
3
È
ef
2 3
√
f 2 3
√
f ′′′
f ′
∼ 3
q
e2f (17)
That means that the accuracy of the symmetrical dierence quotient is be-
tween one or two orders of magnitude better than of the asymmetrical dier-
ence quotient. Therefore h should be also chosen the correct power of ef or em
times a characteristic scale xc. [22]
4.1.5 Advanced methods of numerical approximation
Unfortunately, the simple approaches of dierent types of the dierence quo-
tient do not yield the desired results in respect of accuracy. To achieve better
results, the exploration of the function's behavior and assumptions of smooth-
ness or analyticity are required to get higher-order terms in a Taylor expansion
that have some meaning. The drawback of these methods is the need of mul-
tiple evaluations of the function f .
Richardson's deferred approach to the limit According to the general
approach of Richardson's deferred approach to the limit [24] one seeks to ex-
trapolate h→ 0. The result are nite dierence calculations with smaller and
smaller nite values of h. Using Neville's algorithm, an algorithm for poly-
nomial interpolation, each new nite dierence calculation is on the one hand
used for the extrapolation of higher order and on the other hand for the extrap-
olation of previous lower orders but with a smaller h-value. Implementations
of this algorithm lead to an approximate derivative and an estimation of its
error. [22]
31
4 Geometric Background
Chebyshev polynomial approximation If a fairly smooth function is given
and it will be evaluated at several arbitrarily chosen points then a Chebyshev
polynom could be used to approximate the function within a given interval.
Instead of derivating the function itself, derivates of the Chebyshev polynom
could be used. A Chebyshev polynom Tn(x) of degree n has the explicit formula
Tn(x) = cos(n arccosx) (18)
Details can be found in [22].
4.2 Curve representation
In the following section the parametric form of a curve representation both
in two and in three dimensions will be introduced. Also the concepts of the
parameter transformation and the so-called arc length will be explained. The
knowledge of these things provides the background for the dierential geometry
calculations on curves explained in section 4.3.
4.2.1 2D curve representation
Plane curves are mostly described analytically in a parametric form:
x = x(t), y = y(t) (19)
where x and y are the Cartesian coordinates of the point P (x, y). x(t) and
y(t) are real unique functions of a real parameter t which are continuous in a
shared interval and not constant. The point P (x, y) describes an arc øAB of
a real continuous plane curve c. To exclude some special cases it is further
requested that the mapping between t in its interval and of the points P of
the continuous arc øAB has to be continuous and reversible unique. It is also
required that the functions x = x(t) and y = y(t) have continuous derivatives
relative to t in its closed interval. These two derivatives also must not be zero
at the same time. For further exploration of the curvature the second and third
derivatives of the curve, in some cases up to the n-th derivative, are required.
The coordinates of the curve points P can also be seen as the coordinates of
the position vector x of point P :
x = x(t) = x(t)e1 + y(t)e2 (20)
where e1 and e2 are the unit vectors of the coordinate axes' directions. The
derived position vector x′ is also continuous within its interval and never equal
to zero:
32
4 Geometric Background
x′ = x′(t) = x′(t)e1 + y′(t)e2 6= 0 (21)
Geometrically interpreted x′ is the tangent vector of curve x in point t. The
length of the tangent vector depends on the parametric scale of t and its
direction is the same as the curves'. [29]
Parameter transformation Often the parameter t is replaced by a new fea-
sible parameter s
t = t(s) (22)
where t(s) is a monotone and continuous derivable function relative to s and
dt(s)
ds
6= 0 (23)
If dt
ds
> 0 the orientation of curve c stays the same otherwise the orientation
is swapped. After the parameter transformation the curve c can be written as
[29]
x = x(s) (24)
Arc length The vector dierential
dx(t) =
dx(t)
dt
dt = x′(t)dt (25)
of the smooth curve
x = x(t) = x(t)e1 + y(t)e2 (26)
is parameter invariant relative to t. Its inner square
(dx(t))2 = x′2(t)dt2 (27)
is also invariant relative to movement.
After the parameter transformation t = t(s) and dt
ds
6= 0 the position vector
x(t) becomes x(t(s)) = x(s) and the inner square is
x′2(t)dt2 = x′2(s)ds2 (28)
If for all s of the arc it is imperative that
33
4 Geometric Background
x′2(s) = x′2(s) + y′2(s) ≡ 1 (29)
i.e. the tangent vector has the xed length of 1 then parameter s is special.
s is called the arc length of curve and ds is called the arc element or the arc
dierential.
The square of the arc element is
ds2 = (dx)2 = dx2 + dy2 (30)
The arc length s of the curve between the points t0 and t can be obtained
by the integration of the arc element ds
s =
Z t
t0
È
x′2(t)dt =
Z t
t0
È
x′2(t) + y′2(t)dt = s(t) (31)
The arc element is invariant relative to parameters and to movement. It
is also the dierential of lowest order with these features, i.e. the simplest
dierential invariant of the curve. Through integration the arc length s can
be obtained which is also the simplest integral invariant of the curve. ds is
called the natural dierential and s the natural parameter of the curve. All
further obtained values and vectors based on the natural parameter are also
parameter invariant.
Geometric meaning To obtain the length of an arc c = øAB of the curve x =
x(t) we inscribe the arc c with arbitrary intermediate points Pv = (x(tv), y(tv))
with parameter values ta = t0 < t1 < t2 < · · · < tn−1 < tn = tb = t into a
polyline P = (A,P1, P2, . . . , Pn−1, B) and determine its length:
s(P) =
nX
v=1
|x(tv)− x(tv−1)| (32)
The more intermediate points Pv are used the better is the estimation of the
real length of the arc according to the triangle inequality. If the set of polylines
{s(P)} which can be inscribed into the arc is bounded above the arc c can be
assigned with a nite arc length s = sup s(P) and is called rectied.
Theorem: Each continuous dierentiable curve x = x(t) can be called recti-
ed. Its arc length s = s(t) is a continuous derivable function of t described
with the integral [29]
s = s(t) =
Z t
t0
È
x′2(t)dt =
Z t
t0
|x′(t)|dt (33)
34
4 Geometric Background
4.2.2 3D curve representation
The basic principles for curves in three dimensions are mainly the same as
for curves in 2D, see section 4.2.1. The parametric representation of a three
dimensional curve consists of three in its interval ta < t < tb unique continuous
derivable functions x(t), y(t) and z(t) relative to a feasible parameter t which
derivations are not equal to zero at the same time. These functions determine
as coordinates of the position vector
x = x(t) = x(t)e1 + y(t)e2 + z(t)e3 (34)
a continuous derivable 3D curve.
The tangent vector at a position t is given through the derivation of the
position vector
dx
dt
= x′ = x′(t) = x′(t)e1 + y′(t)e2 + z′(t)e3 (35)
Because of the continuity of the tangent vector x′(t) the curve can be called
rectied and its arc length s is given through the integral
s =
Z t
t0
È
x′2(t)dt =
Z t
t0
È
x′2(t) + y′2(t) + z′2(t)dt = s(t) (36)
The arc element of a 3D curve is also independent of the parameter repre-
sentation x(t) and the spatial position of the curve. The arc length s is called
the natural parameter of the 3D curve and is bounded above by the length of
an inscribed polyline. Choosing the arc length s as parameter of the curve
x = x(s) = x(s)e1 + y(s)e2 + z(s)e3 (37)
then the tangent vector
t = x′ = x′(s) = x′(s)e1 + y′(s)e2 + z′(s)e3 (38)
has a xed length of one [29]
|t| = |t′(s)| = 1 (39)
35
4 Geometric Background
4.3 Dierential geometry on a curve
On the following pages the mathematical and the geometrical basic principles
necessary for understanding the concepts of dierential geometry on a curve
presented in this thesis will be explained. At the beginning the concept of the
Frenet frame will be presented and the terms curvature and torsion will be
dened. In the following subsection the concepts of the center, the circle and
the sphere of curvature will be introduced. Finally the plane of curvature will
be explained.
4.3.1 Frenet frame in curve points
The Frenet frame (see gure 9) is a local coordinate system in each point of
a curve. Using the natural parameter s one can exploit the advantage that
the derivatives x′(s), x′′(s) . . . of a curve x(s) are invariant of movement and
parameter.
Figure 9: This gure shows the Frenet frame in a curve point P . The Frenet
frame consists of three vectors: the tangent vector t, the normal
vector n and the binormal vector b.
For x′′(s) 6= 0 the Frenet frame can be dened as:
t = x′(s) . . . tangent vector
n =
x′′(s)
|x′′(s)|
. . . normal vector
b = t× n . . . binormal vector
(40)
For each natural parameter s x′(s) is a unit vector. From this it follows that
36
4 Geometric Background
x′2 = t2 ≡ 1 (41)
and therefore
x′x′′ = tt′ ≡ 0 (42)
i.e.
t ⊥ n (43)
From this it follows that both unit vectors, the tangent vector t and the
normal vector n, are pairwise orthogonal and form a right-handed coordinate
system.
The naming of the three vectors forming the Frenet frame can be explained
as follows: t is an orientation vector of the tangent at position s of curve x(s).
A straight line orthogonal to a tangent is called a normal and both the normal
vector and the binormal vector fulll this condition.
If the curvature
k(s) = |x′′(s)| (44)
in x(s) exists, there is also a frenet frame. x′′(s) can be seen as a measure
for the variation from a straight-lined run of the curve. If |x′′(s)| = 0 in an
interval the run of the curve is straight-lined in this range. The curvature of a
curve in 2D can also be regarded as a measure how fast the tangent is rotating
around its curve point.
If we want to know the variation of the vectors of the Frenet frame when
moving along the curve we assume the following system of equations:
t′ = a11t + a12n + a13b
n′ = a21t + a22n + a23b
b′ = a31t + a32n + a33b
(45)
Given that t′(s) = x′′(s) = kn(s) we can state that a11 = a13 = 0 and a12 = k.
Because t, n and b form a trihedron, i.e.
t2 = n2 = b2 = 1 (46)
and
tn = nb = bt = 0 (47)
37
4 Geometric Background
we can solve the former equations for aik:
a11 = t′t = 0 a12 = t′n = k a13 = t′b = 0
a21 = n′t a22 = n′n a21 = n′b
a31 = b′t a32 = b′n a33 = b′b
(48)
Concerning
tt = nn = bb = 1 (49)
it is imperative that
t′t + tt′ = 0
n′n + nn′ = 0 (50)
b′b + bb′ = 0
i.e.
tt′ = nn′ = bb′ = 0 (51)
From
tn = nb = bt = 0 (52)
it follows that
t′n + tn′ = 0
n′b + nb′ = 0 (53)
b′t + bt′ = 0
i.e.
a12 = −a21
a23 = −a32 (54)
a13 = −a31
We also state that
w(s) = n′b = nb′ (55)
and call it the torsion at position x(s). The torsion of a curve point can
be regarded as a measure how fast the corresponding plane of curvature is
rotating around the tangent. A curve without torsion lies within its plane of
curvature.
38
4 Geometric Background
From these assumptions follow these dierential equations, the Frenet for-
mulas:
t′ = kn
n′ = −kt + wb (56)
b′ = −wn
The torsion w(s) is a measure for the variation of a planar run of the curve.
If w(s) = 0 in an interval the run of the curve is planar in this range. Because
of b′(s) = −w(s)n(s) the variation of the binormal can be seen as a rotation
around the tangent which at the same time is a rotation of the plane spanned
out of t(s) and n(s). So the torsion w(s) brands the "winding" of the curve
out of this plane.
Example: straight line:
x(s) = x+ sa
where |a| = 1 and s ∈ IR.
1st and 2nd derivation:
x′(s) = a
x′′(s) = 0
So both the curvature and the torsion of a straight line which run is straight-
lined and planar, amount to 0.
k(s) = 0
w(s) = 0
Example: circle:
x(s) = O + r cos
s
r
i + r sin
s
r
j
where r > 0 and s ∈ IR.
1st and 2nd derivation:
x′(s) = − sin
s
r
i + cos
s
r
j
x′′(s) = −1
r
cos
s
r
i− 1
r
sin
s
r
j
From this it follows the curvature of a circle
k(s) = |x′′(s)| = 1
r
is constant and inversely proportional to its radius.
39
4 Geometric Background
The torsion of a circle which is a planar geometric primitive amounts to [15]
w(s) = 0
Example: helix: As for a straight line and a circle a helix has a constant
(independent of parameter s) curvature and torsion:
k(s) =
r
r2 + h2
w(s) =
h
r2 + h2
If w > 0 (or h > 0) it is a left-handed helix and if w < 0(or h < 0) it is a
right-handed helix. [15]
4.3.2 Center, circle and sphere of curvature
Given the parametric representation x = x(s) of a curve.
In the case of k(s0) 6= 0 we look for a circle that approximates the curve in
point x(s0) as good as possible. If we nd the midpoint M we can calculate
the radius r as the distance between the midpoint M and the point x(s0) on
the curve:
r = |v(Mx(s0))| (57)
The circle is coplanar with the plane dened by the midpoint M and the
tangent on the curve in point x(s0).
Given the midpoint M and the radius r we can state the function
f(s) = (v(Mx(s)))2 − r2 (58)
This function can be seen as a measure for the deviation of the distance
between the midpoint M and the point x(s) from the radius r. That is the
deviation of the curve from the sphere S(M, r) with the midpoint M and the
radius r. According to the Taylor expansion the approximation is good if
f(s0) = 0
f (v)(s0) = 0, for as much v as possible
With the Frenet formulas in mind and assuming that m is the position vector
of M and r(s) is the position vector of r(s) we can state:
f(s) = (r(s)−m)2 − r2 (59)
40
4 Geometric Background
f ′(s) = 2(r(s)−m) · r′(s) = 2(r(s)−m) · t (60)
f ′′(s) = 2r′(s) · t + 2(r(s)−m)t′ = 2 + 2k(r(s)−m)n (61)
f ′′′(s) = 2[k′(r(s)−m)n+ktn+k(r(s)−m)(−kt+wb)] = 2(r(s)−m)[−k2t+k′n+kwb]
(62)
So
f(s0) = (r(s0)−m)2 = 0 , i.e. x(s0) ∈ K(M, r) (63)
f ′(s0) = 2(r(s0)−m) · t = 0 , i.e. r(s0)−m⊥t (64)
that is x(s0) is a point on the circle and the radius is perpendicular to the
tangent in x(s0).
If we assume that r(s0) − m = xn(s0) + yb(s0) and insert into equation 61
we get
f ′′(s0) = 2(1 + k(xn + ybn)) = 2(1 + kx) = 0 (65)
i.e.
x = − 1
k(x0)
(66)
Equation 62 leads us to
f ′′′(s0) = 2

−1
k
n + yb

[−k2t + k′n + kwb] = 2

−k
′
k
+ kwy

= 0 (67)
i.e.
y =
8><>:
k′
k2w
for w(s0) 6= 0
arbitary for w(s0) = k′(s0) = 0
(68)
The midpoints of all spheres with f(s0) = f ′(s0) = f ′′(s0) = 0 fulll the
equation
M = x(s0) +
1
k
n− yb (69)
and lie all on a straight line a, called axis of curvature which is parallel to
the binormal of the curve in x(s0). a goes through M(s0) = x(s0) + 1
k
n with
the orientation vector b. On all this spheres lies a circle with midpoint M(s0),
the so-called center of curvature (see gure 10), and radius
41
4 Geometric Background
ρ(s0) =
1
k(s0)
(70)
the so-called radius of curvature.
The midpoint of the circle of curvature (see gure 10):
M(s) = x(s) + ρ(s)n(s) = x(s) + ρ2(s)x′′(s) (71)
Figure 10: This gure shows an illustration of the center Mc, the circle c, the
sphere s and the plane of curvature σ in a curve point P of curve
x(t). The Meusnier point Ms (see section 4.4.4), the center of the
sphere of curvature s, is also shown. The Frenet frame consisting
of the tangent t, the normal n and the binormal b is shown in curve
point P . The amount to move in the direction of the normal for
achieving the center of curvatureMc is labeled as ρn and the amount
in the binormal's direction to get the Meusnier point Ms as βb.
Theorem: The curvature k(s) of curve is equal to the inverse of the radius
of the curvature ρ(s), i.e.
k(s) =
1
ρ(s)
(72)
42
4 Geometric Background
A sphere that touches the curve at at least third order exists if w(s0) 6= 0
or w(s0) = k′(s0) = 0. It is called sphere of curvature (see gure 10) with
midpoint
M = x(s0) +
1
k
n− k′
k2w
b (73)
if w(s0) 6= 0 or
M = x(s0) +
1
k
n− yb (74)
if w(s0) = k′(s0) = 0 and y arbitrary. [15]
4.3.3 Plane of curvature
Given a twice continuous derivable and not straight-lined curve in space
x = x(t) = x(t)e1 + y(t)e2 + z(t)e3 (75)
We choose a xed point t0 on the curve and close-by two movable points t1
and t2. These points determine a plane
σ(t0, t1, t2) = a · y + k = 0 (76)
where a is the normal vector of the plane, y is the position vector of an
arbitrary point on the plane and k is a scalar. The plane of curvature (see
gure 10) in point t0 of the plane is obtained if the movable points t1 and t2
simultaneously and independent of each other tend to the xed point t0. It
follows that the plane of curvature σ(t0) contacts the curve x(t) on position t0
in three points.
If
f(t) = a · x(t) + k = 0, a 6= 0 (77)
is true then the point t lies on the plane σ.
If the curve x(t) is twice continuous derivable then the function f(t) has a
rst and a second derivation f ′(t) and f ′′(t). Because the three points t0, t1
and t2 lie on the plane σ it follows that
f(t0) = 0, f(t1) = 0, f(t2) = 0 (78)
Assuming a dierentiable function f(x) whose derivative is a continuous
function and f(a) = 0 and f(b) = 0 (a < b) then there must be at least one
43
4 Geometric Background
point c between a and b at which f ′(c) = 0. Therefore there has to be a
position t∗01 between t0 and t1 and a position t∗12 between t1 and t2 where
f ′(t∗01) = 0, f ′(t∗12) = 0 (79)
There's also a position t∗∗01 between t
∗
01 and t
∗
12 where
f ′′(t∗∗01) = 0 (80)
If the positions t1 and t2 tend to t0, t1 → t0 and t2 → t0, then t
∗
01 and t∗∗01
also tend to t0. From this it follows that
f(t0) = 0, f ′(t0) = 0, f ′′(t0) = 0 (81)
or
f(t0) = a · x(t0) + k = 0
f ′(t0) = a · x′(t0) + k = 0 (82)
f ′′(t0) = a · x′′(t0) + k = 0
If the cross product at position t0 is not equal to zero
x′(t0)× x′′(t0) 6= 0 (83)
then the normal vector of the plane of curvature σ at position t0 is
a = λ(x′(t0)× x′′(t0)), λ 6= 0 (84)
Because
k = −ax(t0) (85)
and the formula of the normal vector can be cancelled by λ we can state the
equation of the plane of curvature at position t0 as

X − x(t0) x′(t0) x′′(t0)
Y − y(t0) y′(t0) y′′(t0)
Z − z(t0) z′(t0) z′′(t0)
 = 0 (86)
Theorem: The twice continuous derivable non straight-lined curve in space
x(t) has in all points t0, provided that the derived vectors x′ and x′′ are linearly
independent, i.e. x′(t0)× x′′(t0) 6= 0, a uniquely determined plane of curvature
σ(t0). [29]
44
4 Geometric Background
In other words, the plane of curvature is unique for k(s) 6= 0. It is determined
through the point x(s) and its normal vector, the binormal b of the curve in
point x(s). The plane is spanned of the tangent t and the normal n in point
x(s):
σ = x(s) + st + tn (87)
There are another two important planes through point x(s) spanned of the
vectors of the frenet frame, the normal plane and the rectifying plane. The
normal plane is perpendicular to the tangent t of the curve in point x(s) and is
spanned of the normal n and the binormal b in point x(s). The rectifying plane
is orthogonal to the normal n of the curve in point x(s) and is spanned of the
tangent t and the binormal b in point x(s). The normal plane, the rectifying
plane and the plane of curvature are each normal to one another and spanned
of two of the vectors forming the Frenet frame, the tangent, the normal and
the binormal vector of the curve in point x(s). [15]
4.4 Dierential geometry on a surface
As for the curves in section 4.3 some basic mathematical and geometrical
principles necessary for dierential geometry on a surface will be introduced in
this section. At the beginning, the idea of surface curvature and their dierent
types will be explained. After that, the concept of the Frenet frame in surface
points (similar to the Frenet frame in curve points presented in subsection
4.3.1) will be presented. Next, a special point, the Meusnier point, will be
explained. Finally a short explanation of general sweeps is given.
4.4.1 Surface curvature
The curvature of a given point on a surface stands for the variation of the
normal vector in this point when moving the point on the surface. The principle
of curvature is easier explained with curves but can easily be extended to
surfaces.
Figure 11 shows a surface and the surface normal n at a given point P . The
corresponding tangent plane of the surface in that point (which is determined
by the point and the surface normal n) is spanned by two vectors e1 and e2. If
one imagines each of these two vectors, e.g. e1, to span a plane with the surface
normal n this plane is orthogonal to the tangent plane, i.e. it is a normal plane.
The result of the intersection of this normal plane and the surface is a plane
curve, e.g. c1, which lies within this normal plane and goes through point P .
This curve c1 has a specic curvature in point P and this type of curvature is
called normal curvature of the surface.
Of course, there is an innite amount of directions and corresponding in-
tersection curves at a given point on a surface. The two most interesting
45
4 Geometric Background
Figure 11: This gure shows a surface and the surface normal n in surface
point P . The vectors e1 and e2 indicate the principal curvature
directions. Each one of these directions can span a plane with
the surface normal n. The resulting normal planes can further be
intersected with the surface. This results in the plane intersection
curves c1 and c2. The curvatures of these intersection curves in
point P are the maximum and the minimum curvature magnitudes
κ1 and κ2.
directions, the directions of the maximum and the minimum curvature are
called principal curvature directions e1 and e2. The corresponding scalar cur-
vatures are referred to as principal curvature magnitudes κ1 and κ2. As κ1
denotes the maximum curvature κ1 ≥ κ2 is always true. The curvature of a
surface point in any direction can be calculated from the principal curvature
directions. [8]
Surfaces can be regarded as an image of a plane region G, i.e.
x = x(u1, u2) = (x(u1, u2), y(u1, u2), z(u1, u2)) (88)
This parameter representation will be feasible, if and only if it is triply
continuous derivable and the partial derivatives x1 and x2 are not parallel.
If one parameter is constant, the equation is similar to the parameter repre-
sentation of a curve. If we assume parameter u1 constant, we obtain a curve
with a parameter u2 and vice versa. Tangent vectors of these parameter lines
can be determined by partial derivatives of the curve
xi =
∂x(u1, u2)
∂ui
(89)
For a feasible parameter representation it is also mandatory that [15]
x1 × x2 6= 0, ∀(u1, u2) ∈ G (90)
46
4 Geometric Background
4.4.2 Types of surface curvature and special points
Based on their principal curvature, magnitudes κ1 and κ2 surface points and
the corresponding surface curvatures can be classied in the following cate-
gories (see gure 12):
 Elliptical points : κ1, κ2 > 0, κ1 6= κ2, convex curvature, elliptical surface
curvature
 Hyperbolic points : κ1 > 0, κ2 < 0, surface is saddle-shaped, hyperbolical
surface curvature
 Parabolic points : κ1 or κ2 is zero, surface is locally cylinder-shaped,
parabolic surface curvature
 Umbilical points : κ1 = κ2, locally sphere-shaped or planar, curvature di-
rections are not well-dened. Spheres and planes consist only of umbilical
points with constant curvature. [8]
4.4.3 Frenet frame in surface points
Given a feasible parametric representation of surface x = x(u1, u2), an arbitrary
surface point x(s) and an arbitrary surface curve x(t) = x(u1(t), u2(t)) through
point x. The tangent vector
x′ = x1u
′
1 + x2u
′
2 (91)
lies in the plane ε spanned of x1 and x2.
Given a second feasible parametric representation of the same surface with
the parameters u1 and u2, the tangent vectors x1 and x2 of the parametric lines
u1 and u2 through point x(s) lie within the plane ε. That means the plane
ε is independent of the chosen parameter values of the surface and is called
tangent plane.
The cross product
N = N(u1, u2) =
x1 × x2
|x1 × x2|
(92)
is orthogonal to the plane ε and is called the normal vector of the plane.
The triplet
x1, x2,N
is called Frenet frame of the surface.
The Frenet frame is a right-handed system with
47
4 Geometric Background
Figure 12: This gure shows dierent types of surface curvature and their cor-
responding special points (from left to right and from top to bot-
tom) as explained in section 4.4.2 for arbitrary curves on the surface:
a hyperboloid (hyperbolic points), an ellipsoid (elliptical points), a
cylinder (parabolic points) and a sphere (umbilical points). The
Frenet frame in a point of a surface curve is also shown for every
object.
48
4 Geometric Background
x1, x2⊥N (93)
and
N2 = 1 (94)
although generally x1 and x2 are not orthogonal unit vectors.
In general x1 and x2 are dependent of the chosen parameter values of the
surface whereas N is uniquely determined except of the prex. [15]
4.4.4 Meusnier point
Given an arbitrary surface and a point x(s) on this surface the tangent plane
τ in point x(s) approximates the surface in a linear way. Given an arbitrary
tangent t in point x(s) there are countless surface curves with this tangent t.
Each of these curves c has its own circle of curvature. Meusnier has shown that
all these circles of curvature lie on a shared sphere, called Meusnier sphere.
The midpoint of the Meusnier sphere is called Meusnier point (see gure 17).
[10]
4.5 Geometric properties of implemented surface classes
This section explains the geometric properties of the surface classes imple-
mented in Constuct3D for this thesis. At the beginning the general properties
and the concept of a helical transformation will be explained. After that,
the special features of helices (helical transformation of a point) and helical
sweep surfaces (helical transformation of a curve) and their special types will
be introduced.
4.5.1 Helical transformations
A helical transformation is a spatial movement composed of a rotation around
an axis and a proportional translation along that axis. It is imperative that
s = c · σ (95)
where s is the translation distance, σ is the angle of rotation and c 6= 0 is the
helical parameter. If the helical parameter c is equal to zero, it will be only an
ordinary rotation.
The corresponding distance of the translation h = 2cπ to a full revolution
of σ = 2π is called full height of the screw and can be directly measured on
49
4 Geometric Background
a helical object. The helical parameter c will be obtained if the full height of
the screw is known:
c =
h
2π
≈ h
6
− 5% (96)
Assuming that the axis of the screw is the z-axis of a polar coordinate system
(r, φ, z). The screw transformation of a point P0(r0, φ0, z0)→ P (r, φ, z) can be
described as
r = r0σ
φ = φ0 + σ (97)
z = z0 + cσ
Written in Cartesian coordinates where
x = r · cosφy = r · sinφ (98)
for the transformation P0(x0, y0, z0)→ P (x, y, z) the following equations can
be obtained:
x = x0 cosσ − y0 sinσ
y = x0 sinσ − y0 cosσ (99)
z = z0 + cσ
Looking along the axis contrary its orientation two types of helical rotation
can be distinguished: if the rotation is counterclockwise it is a right-handed
helical rotation, c > 0, otherwise a left-handed, c < 0. [34]
Table 1 summarizes surface types that can be generated by a helical trans-
formation.
object to screw helical object
point helix, helical curve
curve helicoid, helicoidal surface, helical sweep surface
straight line ruled helicoid
circle cyclic helicoid
surface enveloping helicoid
plane developable helicoid, helicoidal torse
sphere tubular helicoid
Table 1: Summary of surface types generated by a helical transformation.
50
4 Geometric Background
Helix A helix (see gure 13) is the path of a point subordinated to a helical
transformation. The following parametric representation of a helix with xed
values x0, y0, z0 and a variable angle of rotation σ can be obtained assuming
that the x-axis of the coordinate system contains point P0, i.e. x0 = r and
y0 = z0 = 0:
Figure 13: This gure shows an illustration of a helix in front view. A point
P is rotated around an axis a. The radius is labeled as r and the
full height of the helix is labeled as h.
x = r · cosσ
y = r · sinσ (100)
z = cσ
The equations of y and z can be considered as a the upright projection of
the xz-plane and the projection of the helix is a sine wave
y = r · sin z
c
(101)
Recapitulatory the upright projection of a helix on a plane parallel to the
axis is a sine wave with a full height of h = 2cπ and the ground view on the
xy-plane is a circle with radius r.
The helix has several special features: a helix lies on a coaxial cylinder Γ
with radius r and intersects all generators at the same angle. That means
the helix is a loxodrome of the cylinder. If the cylinder Γ is developed into a
51
4 Geometric Background
plane the helix becomes a straight line, i.e. the helix is also a geodesic of the
cylinder, i.e. the shortest path between two points.
Helical sweep surface A helical sweep surface Φ (see the screenshot in gure
19) is formed when a curve is transformed by a helical motion, i.e. rotated
and simultaneously moved along an axis. The curve c is the generator of the
helical sweep surface Φ. Below some types of helical sweep surfaces will be
described. Details can be found in [2, 34].
There are several special cases of helical sweep surfaces: i.e. ruled helicoids
and cyclic helicoids.
Ruled helicoids A ruled helicoid is generated if a straight line s is screwed
around an axis. A ruled helicoid is called closed when the axis and the straight
line intersect, otherwise it is called open. It is called right when the axis and
the straight line are orthogonal, otherwise it is called skew. Hence the following
cases can be discerned:
 right closed ruled helicoid :
It is generated if a straight line s orthogonal to the axis, that also in-
tersects the axis, is screwed. This type is also called right helicoid. The
stages of a spiral staircase lie on such a type of helical surface.
 right open ruled helicoid :
This surface is generated if a straight line s orthogonal to the axis, but
not intersecting the axis, is screwed.
 skew closed ruled helicoid :
This helical sweep surface is generated if a straight line s that intersects
the axis in a skew angle γ is screwed.
 skew open ruled helicoid :
This is the most general type of a helicoid. A straight line s that crosses
the axis in a skew angle γ is screwed.
 developable helicoid :
A developable helicoid consists of all tangents of a helix.
Circle helicoids If a circle c is screwed a circle helicoid is generated. Depen-
dent of the position of the circle relative to the axis there exist several dierent
cases:
 circle helicoid :
The plane of the circle c is normal to the axis. It is only necessary to
specify the helix m of circle's midpoint and the radius of the circle. All
points of the circle create helices congruent to m. A circle helicoid can
52
4 Geometric Background
also be created if the circle is swept along the helix m and therefore it is
also a sweep surface.
 meridian circle helicoid :
This type of helicoid is generated if the circle's plane lies in the same
plane as the axis.
 tubular helicoid :
If a sphere (assumed its midpoint lies not on the axis) is screwed, a surface
will be generated that is both a helical surface and a tubular surface.
The mid line of the tubular helicoid is the helix of the midpoint of the
sphere. The surface of the tubular helicoid is touched by a great circle of
the sphere whose plane is perpendicular to the mid line. Therefore this
tubular helicoid can also be generated by this circle. [34]
4.5.2 General sweeps
For generating a general sweep surface (or translation surface) a prole curve p
is moved along a second curve, the leading curve l. In most cases both curves
share a point O. The prole and the leading curve may both be two and three
dimensional curves.
For instance could a cylinder not only be seen as a straight line rotated
around an straight-lined axis but also as a straight line (the prole curve)
moved along a circle (the leading curve) perpendicular to the prole curve (see
the screenshot in gure 20). The prole and the leading curve can also always
change their roles. The cylinder can also be generated if the circle (the prole
curve) is moved along a straight line (the leading curve). [34]
53
5 Design
5 Design
This section will give an overview over the design ideas which have been imple-
mented in the course of this thesis. First the start up process of Construct3D
is described shortly. After that the functions which have been implemented are
described generally together with their necessary Construct3D input elements
and the resulting output elements. Finally, the extension of the user interface
which became inevitable for integrating the new functions is mentioned.
5.1 Construct3D startup
Construct3D [14] is a 3D dynamic geometry construction tool and it is based
on the Augmented Reality System Studierstube [26]. For further details of the
general concept of Construct3D see section 3.3.
At its start up Construct3D initializes a 3D window with the largest possible
size to create the illusion of covering the whole 3D space. After that the
user interface is initialized, the personal interaction panel (PIP), a hand-held
tracked tablet with the projection of a menu system on it. New points can be
created by turning on the point mode on the PIP and pointing in the 3D space
and pressing a button on a tracked pen. After turning o the point mode
points and previously created objects can be selected with the pen.
5.2 Workow for creating a new object
To create a new object in Construct3D rst of all the user has to select the
required input elements. If a user does not know which input elements are
required this information will be found in the Help notes box positioned on
the top of the PIP. By moving the pen over a construction menu item a help
text is shown. For example, to create a circle of curvature, a curve and a point
on this curve have to be selected. Points can be created when the point mode
of Construct3D is turned on by clicking on the SoToggleButton Point on the
right of the PIP with the aid of the pen (see gure 21). Then, by clicking the
pen's button, a point in space will be created. After that the point mode has
to be turned o again. Now the object or point closest to the tracked pen will
be highlighted (the point gets overlaid with a white wireframe structure) and
can be selected by clicking the pen's button. When creating a Points on curve
curve the resulting curve will go exactly through the control points. Therefore
the points have to be selected in the correct order. After that the sub menu
3D in the PIP's construction menu has to be selected. By clicking on the
SoPushButton Points on curve a curve through the previously selected point
is generated (a SoPushButton is the implementation of an ordinary button
which goes up and down and forces the immediate execution of a command).
54
5 Design
Now this new curve has to be selected. After turning on the point mode again
a point can be positioned on the curve by clicking close by the selected curve.
The new point will now be positioned on the selected curve. The point mode
has to be turned o again to be able to select the curve and the point on the
curve. Now the Digeo sub menu must be chosen. At one time there are only
four of six sub menus visible. If the Digeo sub menu is not visible in the
construction menu bar the user will have to click on one of the little arrows
left of the construction menu bar for switching between the sub menus. From
the Digeo sub menu the Circle of curvature button ought to be chosen. If
the user moves the pen over the button without clicking he will get a preview
of the circle in a dotted line style. Finally, by clicking the button, the circle of
curvature will be generated (see gure 16).
5.3 Dierential geometry functions
This section describes shortly the general concepts of the newly implemented
functions for dierental geometry. To create an object in Construct3D the
input elements have to be selected rst. For every function the required input
elements and the resulting output element will be given.
5.3.1 Frenet frame
Figure 14: This gure shows a screenshot of a Frenet frame in Construct3D.
The input elements, a curve and a point on this curve, are colored
in red. The output element is the Frenet frame in the previously
selected curve point consisting of the tangent, the normal and the
binormal of the curve in that point.
The Frenet frame is a tool for the exploration of curves. It is a coordinate
system that can be "glued" to a point on a curve. If the point on the curve
is moved the Frenet frame will also move and adapt its orientation automati-
cally. Using the natural parameter s one can exploit the advantage that also
55
5 Design
the derivatives x′(s), x′′(s) . . . of a curve x(s) are invariant of movement and
parameter. For further details see sections 4.3.1 and 4.4.3.
Input : To create a Frenet frame (see the screenshot in gure 14) in a curve
point in Construct3D, a curve and one or more points on this curve have to
be selected.
Output : The Frenet frame is displayed in all previously selected curve points.
5.3.2 Plane of curvature
The plane of curvature in a curve point P is a plane that approximates this
curve x(s) best. It is spanned out of the tangent t and the normal vector n of
curve x(s) in point P . Its normal vector is the binormal vector b of curve x(s).
For further details see section 4.3.3.
Figure 15: This gure shows a screenshot of the plane of curvature in Con-
struct3D. The input elements, a curve and a point on this curve,
are colored in red. The output element is the plane of curvature in
the previously selected point.
Input : To create the plane of curvature (see the screenshot in gure 15) in
a curve point in Construct3D a curve and a point on this curve have to be
selected.
Output : The plane of curvature is displayed in the previously selected curve
point.
56
5 Design
5.3.3 Center and circle of curvature
The circle of curvature is the circle which approximates a curve x(s) in a curve
point P best. The circle of curvature's tangent in point P matches exactly
the curve's tangent in point P . Its radius, the radius of curvature, is the
reciprocal value of the curve's curvature in point P . The center of curvature is
the midpoint of the circle of curvature. Because of the variation of the curve's
curvature, the circle of curvature approximates the curve only in a very small
neighbourhood of point P . For further details see section 4.3.2.
Figure 16: This gure shows a screenshot of the center and the circle of cur-
vature in Construct3D. The input elements, a curve and a point on
this curve, are colored in red. The output elements are the center
and the circle of curvature in the previously selected point. The
center and the circle of curvature are only displayed in one gure to
illustrate the relationship between them. The center and the circle
of curvature have to be constructed independently.
Input : To create the center or the circle of curvature (see the screenshot in
gure 16) in a curve point in Construct3D a curve and a point on this curve
have to be selected.
Output : The center or the circle of curvature are displayed in the previously
selected curve point.
5.3.4 Meusnier point
Given an arbitrary surface and a point P on this surface. Given an arbitrary
tangent t in point P there are countless surface curves with this tangent. Each
of these curves has its own circle of curvature. All these circles of curvature
lie on a shared sphere, the Meusnier sphere. The midpoint of the Meusnier
sphere is the so-called Meusnier point. For further details see section 4.4.4.
57
5 Design
Figure 17: This gure shows a screenshot of the Meusnier point in Con-
struct3D. The input elements, a curve and a point on this curve,
are colored in red. The output element is the Meusnier point of the
previously selected point.
Input : To create the Meusnier point (see the screenshot in gure 17) of
a curve point in Construct3D a curve and a point on this curve have to be
selected.
Output : The Meusnier point is displayed for the previously selected curve
point.
58
5 Design
5.4 Sweep functions
This section describes the general concepts of the implemented functions for
sweep objects, i.e. helical and general sweeps. Again the input elements have
to be selected rst. For every function the required input elements and the
resulting output element(s) will be declared.
5.4.1 Helical sweeps
A helical transformation is a spatial movement composed of a rotation around,
and a proportional translation along, that axis.
Helix A helix is generated when a point P is rotated around, and propor-
tionally moved along, an axis.
Figure 18: This gure shows a screenshot of a helix in Construct3D. The input
elements, an axis (a straight line) and a point, are colored in red.
The output element is the helix. On the left a preview of the helix
in a dotted line style can be seen and on the right the generated
helix.
Input : To create a helix in Construct3D (see the screenshot in gure 18) an
axis (a straight line) and a point have to be selected.
Output : Output is the helix generated through a rotation around and a
translation of a point along an axis.
Helical sweep surface A helical sweep surface is formed when a curve is
transformed by a helical motion, i.e. rotated and simultaneously moved along
an axis.
59
5 Design
Figure 19: This gure shows a screenshot of a helical sweep surface in Con-
struct3D. The input elements, an axis (a straight line) and a curve,
are colored in red. The output element is the helical sweep surface.
On the left a preview of the helix in wireframe can be seen and on
the right the generated helical sweep surface.
Input : To create a helical sweep surface in Construct3D (see the screenshot
in gure 19) an axis (a straight line) and a curve have to be selected.
Output : Output is the helical sweep surface generated through a rotation
around and a translation of a curve along an axis.
60
5 Design
5.4.2 General sweeps
For generating a general sweep surface a prole curve p is moved along a second
curve, the leading curve l. The prole and the leading curve may both be two
and three dimensional curves.
Figure 20: This gure shows a screenshot of general sweep surfaces in Con-
struct3D. The input elements (colored in red) are in both cases a
straight line and a circle. On the left the leading curve is the straigt
line (1) and the circle is the prole curve (2). On the right leading
and prole curve have changed their roles. The result is in both
cases a cylinder (see section 4.5.2).
Input : To create a general sweep surface in Construct3D (see the screenshot
in gure 20) an axis (a straight line) and one or more prole curves (straight
lines or curves) have to be selected.
Output : Output are the general sweep surfaces created through a translation
of the prole curves along the leading curve.
5.5 The Personal Interaction Panel (PIP)
The following section describes the extensions of the user interface which be-
came necessary due to the introduction of the new construction menu item
Digeo. Afterwards a short description of working with Construct3D in a
desktop setup is given.
5.5.1 Extension of the user interface
To integrate the new functions in the existing layout of the personal interaction
panel (PIP) new menu items had to be inserted (see gure 21).
61
5 Design
Figure 21: Screenshot of the Construct3D Digeo menu: it shows the submenu
items of the new digeo functions. In the help notes eld at the
top the objects which have to be selected for the creation of the
requested object can be seen. At the left from the construction
menu there are two arrows for cycling through all the construction
menu's options.
Figure 22: This gure illustrates the scrolling function for the PIP's construc-
tion menu items. Every new construction menu bar stands for a
click on the left arrow button. After clicking the left arrow button
the construction menu items shift one step to the right. Respec-
tively one new menu item appears from the left and the rightmost
menu item disappears.
62
5 Design
To maintain the balanced layout of the PIP relative to size and position of
the displayed items the new menu button Digeo (abbrevation for dierential
geometry) should not just be positioned next to the other menu items. Instead
a scrolling function for the menu items was choosen (see gure 22). Two arrow
buttons left to the construction menu items can be used to shift the menu's
items left or right. After the last menu item the rst menu item is displayed
again, i.e. the menu items build a cycle.
See section 3.3 and [18] for further general details on the PIP. See section
6.6 for details on the implementation of the changes described above.
5.5.2 Keyboard shortcuts for the desktop setup
For quick testing during the implementation process working with Con-
struct3D's desktop setup is the most ecient way. In this setup no extra
hardware like a head mounted display, a tablet, a pen or a tracking system is
necessary. The user works with a conventional personal computer displaying
desktop virtual reality. The mouse can only be used to change the viewpoint.
For moving the PIP and the pen the keyboard must be used (via the cursor
keys and page up and page down). To simulate some of the possibilities of a
setup with a freely movable PIP and a pen some keyboard shortcuts can be
used (see table 2).
shortcut function
home view point returns to home position showing the PIP
end view point moves backwards showing the PIP and the
coordinate system
0 moving the pen
1 moving the PIP
w pen returns to home position
e counter-clockwise rotation around x-axis
d clockwise rotation around x-axis
g counter-clockwise rotation around y-axis
t clockwise rotation around y-axis
r counter-clockwise rotation around z-axis
f clockwise rotation around z-axis
Table 2: Some useful keyboard shortcuts for the desktop setup. The speci-
cations are for a right-handed three-dimensional coordinate system
with the y-axis pointing away from the user.
63
6 Implementation
6 Implementation
This section explains the general class structure of Construct3D. Besides it
will show how the new functions for dierential geometry (center, circle and
plane of curvature, Meusnier point, Frenet frame) and sweep objects (helical
and general sweep objects) have been integrated into existing classes. Then
the program ow of Construct3D during the execution of a new function will
be explained. Finally the integration of new menu elements in the Personal
Interaction Panel will be described.
6.1 Class overview
The main class of Construct3D (see class overview in gure 23) is the user inter-
face class C3D. Its super class is the Studierstube node kit class SoContextKit
which contains the application functionality. It also provides the window and
the PIP sheet geometry. The user interface class C3D is responsible for the con-
nection of the PIP's widgets to methods for object creation. To handle incom-
ing events eciently C3D also inherits methods of Studierstube's Base3D class.
Geometric objects are initialized by methods of C3D such as addPoint() or
addCurve(). These methods create objects of subclasses of the Object3DKit.
Object3DKit is the super class for all geometric objects in Construct3D. To
enable the dragging of the objects Object3DKit is derived from Studierstube's
SoDragKit. For event handling purposes regarding the drag and drop of the
geometric objects SoDragKit is derived from Studierstube's Base3D class. It
is also derived from the class SoBaseKit, the toplevel super class for node kits.
All geometric objects in Construct3D are derived from Object3DKit:
SoPointKit, SoLineKit, SoPlaneKit, SoCubeKit, SoCurveKit,
SoSurfaceKit, SoSphereKit, SoCylinderKit, SoConeKit, SoBoolKit,
SoIntersectionKit and SoTextKit. Object3DKit contains general func-
tionality needed by all geometric objects of Construct3D such as setting the
selection and highlighting state, assigning of colors and materials (dened
in MaterialConstants), determining an object's layer, recording the user
identity and object deletion. Object3DKit has also a method for detecting
a user's wish for dragging an object or for the generation of xed objects
that cannot be dragged. General constants for Construct3D are stored in
C3DConstants.
6.2 Object creation
To create new objects buttons on the PIP have to be pressed and in suc-
cession the corresponding functions for adding a new object in the function
findButtonMethod() are called. For the creation of the dierential geometry
64
6 Implementation
Figure 23: Class hierarchy of Construct3D: At the bottom of the class di-
agram the classes SoPointKit, SoPlaneKit, SoCurveKit and
SoSurfaceKit are shown, augmented with the newly added func-
tions for dierential geometry and helical and general sweep func-
tionality.
tools a curve and a point on that curve have to be selected. For the helical
sweep objects a straight line (the axis) and another straight line or curve (the
object to be swept) and for the general sweep objects at least one object to be
swept must be selected.
In the body of such a function, e.g. addCenterOfCurvature() the number
of selected objects is calculated. If enough objects, in most cases at least
two objects, have been selected a new Kit object is generated, for example
a SoPointKit. If the object creation was successful, properties like the draw
style will be set. After that the new object is added to the scene graph.
The classes SoPointKit, SoPlaneKit, SoCurveKit and SoPlaneKit are sub-
classes of the general class Object3dKit. Further important class dependencies
can be seen in gure 23.
For the implementation of the tools for dierential geometry the classes
65
6 Implementation
SoPointKit, SoCurveKit and SoPlaneKit have been extended by functions
for the calculation of the center, the circle and the plane of curvature as well
as the Meusnier point and the Frenet frame. The geometrical theory of these
concepts is described in section 4. The Frenet frame was implemented as a tool
which can be switched on and o for previously selected points by pressing a
specic button.
For the creation of helical and general sweep objects the classes SoCurveKit
and SoSurfaceKit have been augmented by functions for the computation of
helices, helical sweep surfaces and general sweep surfaces.
6.3 Dierential geometry functions
6.3.1 Constructor and destructor
In all extended classes further constructors were added. A new parameter, the
object type as an enumeration, was added to distinguish between the dierent
types of objects to be generated.
Within the constructor some properties of the object to be generated like
size, name, number, material and behaviour relative to translation and rotation
are set. In the next step it is checked if enough objects of the correct type
have been selected.
For the calculation of the dierential functions always a curve and at least
one point on that curve have to be selected. If the selected point does not lie
on the curve the calculation will be aborted and a warning message will be sent
to the console output. Otherwise the object type is saved in an enumeration
and the names of the objects are stored in objects of the type SoSFName.
For the helical objects a sweep object and an object to sweep around have
to be selected. In succession several other functions refreshing the object lists
are called.
The functions CreateSensors() and setUpConnections() are called to cre-
ate and attach sensors which will react to every change applied to the underly-
ing objects and result in a recalculation of the dependent objects. Afterwards
the function UpdateAcisObject() is called where a function for generating
the object corresponding to the object type is called.
In the destructor the previously generated sensors are detached and nally
deleted.
6.3.2 Common precalculations
For the center, the circle and the plane of curvature, the Meusnier point and the
Frenet frame the same precalculations have to be made. After the initialization
66
6 Implementation
of variables necessary for the following calculations, the selected curve and the
point on the curve are obtained with help of their object names.
There are two types of curves which have to be treated separately. Ordinary
curves and curves resulting from an intersection have a dierent structure in
their boundary representation (see 3.4.1) and are therefore stored as an EDGE
(a_Curve) or a BODY (a_intCurve).
Then the actual position of the point and the corresponding parameter value
on the curve are obtained. Using the ACIS function eval() that takes the
parameter and the position of the point as parameters, the rst and the second
derivation of the curve at the given position can be calculated. From the
cross product out of the rst and the second derivation the binormal vector is
computed. Because the second derivation is not necessarily orthogonal both to
the tangent vector (the rst derivation) and the binormal vector, the normal
vector is calculated as the cross product out of the rst derivation and the
binormal vector.
The curvature of the curve at that position is calculated as the length of the
binormal vector divided by the length of the tangent vector to the power of
three
curvature =
|binormal vector|
|tangent vector|3
(102)
6.3.3 SoPointKit
The class SoPointKit was extended by several functions for the calculation
of the Frenet frame (section 4.3.1), the center of curvature (section 4.3.2) and
the Meusner point (section 4.4.4).
CalCenterOfCurvature() As stated in section 4.3.2 the radius of a circle is
the reciprocal value of its curvature. The midpoint of the circle of curvature,
the center of curvature, is obtained as the point on the curve is moved in
direction of the normal vector by the amount of the curvature radius
center of curvature = point + (norm(normal vector) ∗ curvature radius)
(103)
Finally the position of the new point has to be converted into coordinates
for the point.
CalMeusnierPoint() For the calculation of the Meusnier point the rst
three derivations are needed. These can be obtained with the ACIS function
67
6 Implementation
evaluation() that takes the parameter and position of the point as param-
eters. If the third derivation is equal to zero the Meusnier point cannot be
calculated and an error message is returned.
After the calculation of the curvature also the torsion has to be computed.
First the torsion determinant must be specied where the colums are the vec-
tors of the rst three derivations. To compute the torsion itself the torsion
determinant is divided by the square of the length of the binormal vector.
torsion =
torsion determinant
|binormal vector|2
(104)
If the determinant of the torsion is equal or nearly equal to zero an error
message is sent to the console output because in that case no Meusnier point
exists.
In the next step the derivation of the curvature expression 102 has to be
calculated. The Meusnier point is obtained as the center of curvature is moved
again in direction of the binormal vector by the amount of derived curvature
radius divided by the torsion
Meusnier point = point +
(norm(normal vector) ∗ curvature radius ) + (105)
norm(binormal vector) *
deriv(curvature radius)
torsion

Finally the position of the point has to be converted in coordinates.
CalFrenetFrame() The function CalFrenetFrame() takes a curve as param-
eter because a point can be part of more than one curve. After the initializa-
tion of the variables the rotation node of class type SoRotation for the Frenet
frame is fetched. As stated above the derivations at the given curve position
are calculated. Then the rotation eld of the rotation node is set according to
the directions of the calculated derivations. Therefore the attached node with
a set of tiny coordinate axes is rotated accordingly. Finally the Frenet frame
is set visible.
DelFrenetFrame() The purpose of the function DelFrenetFrame() is to set
the previously calculated Frenet frame of this point invisible.
6.3.4 SoCurveKit
The class SoCurveKit was extended by a function for the calculation of the
circle of curvature (see section 4.3.2).
68
6 Implementation
createCircleOfCurvature() Because the center of curvature is the midpoint
of the circle of curvature the same calculations as for the center of curvature
have to be made (see section 6.3.3). After obtaining the position of the center of
curvature an ellipse is created with the ACIS function api_mk_ed_ellipse().
The function takes amongst others the position of the midpoint, the normal
vector of the circle's carrier surface and the distance between the midpoint
and the point on the curve (the radius). If no circle can be created, an error
message will be sent to the console output.
6.3.5 SoPlaneKit
The class SoPlaneKit has been augmented with the function
calPlaneOfCurvature() for the calculation of the plane of curvature
at a given point on a curve.
calPlaneOfCurvature() After obtaining the position and the parameter of
the point on the curve the rst and second derivation of the curve at that
position are calculated via the ACIS function eval(). Here it is only neces-
sary to calculate the binormal vector from the cross product of the rst and
second derivation. With the aid of a Construct3D intern macro which takes
the position of a point on the plane and the normal vector of the plane (the
binormal vector at that curve point) as parameters a plane is created. This
generic plane has to be converted into a plane of a predened size. Finally
the plane has to be translated to center it around the point on the curve by
manipulating the vertex properties of the plane's construction points.
6.4 Sweep functions
6.4.1 SoCurveKit
The class SoCurveKit was extended by the function createHelicalSweep-
Curve() in order to construct helices (4.5.1).
createHelicalSweepCurve() For the calculation of a helix some parameters
have to be set as the helical parameter and the angle of rotation. At the
moment these parameters are static but Construct3D can easily be adapted
to dynamically change these parameters through the user interface.
First it is checked whether enough suitable objects were selected, in this case
a straight line (the axis) and a point. If the rotational angle has been dened
in degrees because this measure is easier to understand for the user, it has to
be converted in polar coordinates.
69
6 Implementation
In Construct3D straight lines are dened through two points and have there-
fore a dened length. Only straight lines resulting from a calculation, e.g. a
straight line normal to a plane, have an innite length. To ensure that the
helix has the user dened height and does not end when the axis ends the
straight line is explicitly extended to innity.
With the ACIS function api_edge_helix() which takes as arguments the
closest point to the point to screw on the axis, the endpoint of the helix' axis,
the vector from the point to screw (the start point of the helix) and the start
point of the helix' axis and also its distance and the handiness of the helix. If
an error occures during the calculation, an error message will again be sent to
the console output.
6.4.2 SoSurfaceKit
The class SoSurfaceKit was extended by two similar types of sweep surfaces,
a helical sweep surface (function BuildHelicalSweepSurface(), see section
4.5.1) and "general" sweep surfaces (function BuildGeneralSweepSurface()).
For the creation of a sweep there exists a general function in ACIS
api_sweep_with_options(). So-called sweep options have to be set to specify
the properties of the resulting sweep surface like the behaviour at self inter-
section and whether the surface has to be solid.
BuildHelicalSweepSurface() Basically there is no great dierence between
the precalculations for a helix and a helical sweep surface. Also the helical
parameter and the rotational angle have to be specied. Unlike to the helix
where the order of the selected objects was of no relevance for a helical sweep
object the order of the selected objects is important for providing the correct
parameters to ACIS' sweeping function. At the moment the axis has to be
selected rst and the curve that shall be swept second. Then an arbitrary
point from the curve is taken and the helix of this point around the axis is
calculated. The ACIS function api_sweep_with_options() takes the object
to sweep and the helix where the object is swept along as parameters. Finally if
the calculation of the helical sweep surface was successful the resulting surface
has to be triangulated. In case of an error an error message is displayed.
BuildGeneralSweepSurface() The ACIS function api_sweep_with_-
options() is a general function which takes an object to sweep and a path
to sweep along as parameters. In case of the helical sweep curve the path to
sweep along is a helix. Therefore it is easy to generalize this function to a
general sweep function. At least two objects have to be selected. The rst
selected object, a straight line or a curve, becomes the path to sweep along
and all following selected objects are swept along this path.
70
6 Implementation
6.5 Undo/redo list
To provide an undo/redo functionality for every generated object an en-
try is stored in a special node kit, the so-called UndoRedoListKit. The
UndoRedoListKit also provides a eld containing the current position in the
undo/redo list. For a preview of an object only a preview is shown and no
entry in the undo/redo list is made. As a consequence a form of history is
generated. If an inventor le is loaded at the startup of Construct3D only the
commands in the undo/redo list will be processed. If an object is deleted the
corresponding entry in the list will also be deleted. For an undo or a redo
the counter in the UndoRedoListKit has to be decremented or incremented.
Further details see in [5].
6.6 Extension of the user interface
The following code fragment of the Open Inventor le dening the PIP's lay-
out (see section 3.2.2) shows the Open Inventor item hierarchy of the con-
struction bar. After the denition of a Separator for the construction bar
a SoWidgetLayoutGroup with attributes like width, depth, height and num-
ber of colums is dened to which the following items will be added. The
buttons for shifting the menu items to the side are SoPushButtons. Some
attributes like the used texture which is dened in a separate Open Inventor
le only for textures are set. For the realization of the shifting functional-
ity a Switch is used. All possible combinations of menu items are dened in
SoWidgetLayoutGroups. The menu items itself are of type SoToggleButton
because the visibility of the sub menu items is turned on and o every time
the button is pushed. Already dened SoToggleButtons can be reused via the
keyword USE.
DEF CONSTR_BAR Separator {
SoWidgetLayoutGroup {
width 0.036 depth 0.01 height 0.003
numOfCols 2
...
elements NodeKitListPart {
containerNode Group {
DEF CONSTR_LEFT_BUTTON SoPushButton {
onGeometry Separator {
USE ON_GROUP USE
CONSTR_LEFT_BUTTON_TEX
USE LABEL_GEOMETRY }
...
statusBoxText "65"
71
6 Implementation
}
DEF CONSTR_RIGHT_BUTTON SoPushButton {...}
}
}
}
...
DEF CONSTR_SWITCH Switch {
...
SoWidgetLayoutGroup {
...
elements NodeKitListPart {
containerNode Group {
DEF DIFFGEO SoToggleButton {...}
USE TRANSFORM
USE INTERSECT
USE CONSTRUCT
...
The items of the Digeo menu are dened in another Separator. Be-
cause there are submenu items for every menu item of the construction menu
again a Switch is needed. The buttons for the dierential geometry func-
tions are SoPushButtons except for the Frenet frame which is from type
SoToggleButton because the display of the Frenet frame can be switched on
and o.
DEF WORK_SPACE Separator {
DEF WORK_SPACE_SWITCH Switch {
DEF DIFFGEO_PART Separator {
SoWidgetLayoutGroup {
elements NodeKitListPart {
containerNode Group {
DEF CURVATURE_PLANE SoPushButton {...}
DEF CURVATURE_CENTER SoPushButton {...}
DEF CURVATURE_CIRCLE SoPushButton {...}
DEF MEUSNIER_POINT SoPushButton {...}
DEF FRENET_FRAME SoToggleButton {...}
...
The "Transform" menu was augmented with two buttons for the generation
of a helical and a general sweep (see gure 24). The Transform menu has also
72
6 Implementation
Figure 24: Screenshot of the Construct3D Transform menu: At the bottom the
two options for the general and the helical sweep have been added.
The prerequisites for a helical object can be seen at the top in the
help notes eld. At the bottom there are the elds prepared for the
dynamic change of helical parameter and angle of rotation.
been prepared for the dynamic change of the helical parameter and the angle
of rotation of a helical sweep.
The following code fragment shows the specication of a Separator for the
help notes which denes the text shown if the pen moves over the Digeo
menu button. Translation sets the position of the element depending on the
elements higher in the hierarchy. Also the color of the font, the material and
the text to be shown can be specied.
DEF SHOW_HELPNOTES Switch {
...
Separator {
Translation { translation 0 0.016 0 }
USE FONT_COLOR AsciiText
{ justification LEFT string ["Diffgeo"] }
Translation { translation 0 -0.00974 0 }
Material { diffuseColor 0 0 0 }
AsciiText { justification LEFT spacing 1.12985 string
["Opens the Diffgeo Menu -",
"actions in this menu require that",
"objects be selected first."] }
...
73
6 Implementation
6.6.1 Routing of user interface events
The following section explains the denitions for the behaviour of the user
interface scripted in Open Inventor.
Displaying the digeo menu First of all a SoConditionalTrigger for the
digeo menu is dened. If the trigger's comparison evaluates to true the trigger
eld res.
DEF DIFFGEO_TRIGGER SoConditionalTrigger {
boolIn = USE DIFFGEO.on triggerBool TRUE token "0" }
Next the SoFanIn, an engine which is used to connect a number of elds to
a single eld, is augmented with the trigger for the digeo menu.
DEF CONSTR_FAN SoFanIn {
type MFString
in0 = USE DIFFGEO_TRIGGER.tokenOut
in1 = USE TRANSFORM_TRIGGER.tokenOut
in2 = USE INTERSECT_TRIGGER.tokenOut
in3 = USE CONSTRUCT_TRIGGER.tokenOut
in4 = USE THREE_D_TRIGGER.tokenOut
in5 = USE TWO_D_TRIGGER.tokenOut
in6 = USE CONSTR_NONE_TRIGGER.tokenOut
}
If another trigger for the digeo menu is red the digeo state will be set
on.
DEF DIFFGEO_STATE_ON SoConditionalTrigger {
intIn -1 = USE CONSTR_FAN.out triggerInt 0
}
Finally routes are created via SoRoute between the previously dened
elds. The output of the CONSTR_FAN eld determines the child of the
switch WORK_SPACE_SWITCH. If DIFFGEO_STATE_ON evaluates to true the digeo
SoToggleButton is switched on.
SoRoute {
from "CONSTR_FAN.out" to "WORK_SPACE_SWITCH.whichChild"
}
SoRoute {
from "DIFFGEO_STATE_ON.boolOut" to "DIFFGEO.onIn"
}
74
6 Implementation
Scrolling the menu items First two counters are dened via SoCounter
which are engines that count from a specied minimum to a specied maximum
value with a given step size each time a specied trigger res. Because it is
more intuitive the construction menu's items shall shift to the left if the right
button is pressed and vice versa.
DEF NUM_CONSTR_LEFT SoCounter {
min 0 max 5
step 1
trigger = USE CONSTR_RIGHT_BUTTON.triggerOut
reset 0
}
DEF NUM_CONSTR_RIGHT SoCounter {
min 0 max 5
step 1
trigger = USE CONSTR_LEFT_BUTTON.triggerOut
reset 0
}
In the next step it is calculated which items of the construction menu are
currently visible with the aid of a SoCalculator.
DEF CUR_CONSTR SoCalculator {
a = USE NUM_CONSTR_RIGHT.output
b = USE NUM_CONSTR_LEFT.output
expression ["oa=a-b+6", "ob=oa\%6"]
}
Dependent of the SoCalculator's output a route is dened from the result
of the calculator to the construction menu's switch. As stated above, sev-
eral dierent cases for the display of the construction menu's items have been
dened and the result of the calculation routes to the correct menu items.
SoRoute {
from "CUR_CONSTR.ob" to "CONSTR_SWITCH.whichChild"
}
75
7 Results
7 Results
This concluding section shall give a short and, of course, only fragmentary
survey of the possiblities of the newly implemented functions. The following
screenshots are only simple presentations of the new functions. Hopefully the
wide potential of the novel functions will be explored in future learning sessions.
7.1 Dierential geometry functions
7.1.1 Frenet frame
The Frenet frame, a tool for understanding the meaning of curvature, can be
displayed for every point on an arbitrary curve. Figure 25 shows the Frenet
frame for an arbitrary 3D curve. The Frenet frame adapts dynamically when
the curve or the point is changed.
Figure 25: This gure shows the Frenet frame in a point on an arbitrary 3D
curve.
76
7 Results
7.1.2 Plane of curvature
The plane of curvature in a curve point is a plane that approximates this curve
best. It is spanned out of the tangent and the normal vector of the curve in
a point on a curve. Figure 26 shows the plane of curvature of a given curve
point on an arbitrary 3D curve.
Figure 26: This gure shows the plane of curvature in a curve point on an
arbitrary 3D curve.
77
7 Results
7.1.3 Center and circle of curvature
Like the plane of curvature the circle of curvature is a basic form which ap-
proximates the curvature of a curve in a given curve point best. Figure 27
shows the circle of curvature in a given point on the intersection curve of two
cylinders. The center of curvature is basically the midpoint of the circle of
curvature.
Figure 27: This gure shows the center and the circle of curvature in a curve
point on the intersection curve of two cylinders.
78
7 Results
7.1.4 Meusnier point
As explained in section 4.4.4 the Meusnier point is a special point. It is the
midpoint of a sphere that contains all circles of curvature which pass through a
specic point with a given xed curve tangent. Figure 28 shows the Meusnier
point and the Meusnier sphere for a given point on a 3D curve.
Figure 28: This gure shows the Meusnier point and the Meusnier sphere for
a curve point on a 3D curve.
79
7 Results
7.2 Sweep functions
7.2.1 Helical sweeps
Construct3D was augmented with functions to build helical sweep surfaces.
These are sweeps where the sweep object is not only rotated around the sweep
axis but also translated along that axis during the rotation. Sweep object can
be a point or an arbitrary curve.
Ruled helicoids There are dierent forms of ruled helicoids (see section
4.5.1). A right closed ruled helicoid (see gure 29) is generated if a straight
line normal to the sweep axis that also intersects the sweep axis is screwed.
The stages of a spiral staircase lie on such a type of helicoid.
Figure 29: This gure shows a helical sweep surface with a straight line both
as sweep axis and as sweep object.
80
7 Results
Circle helicoids A circle that lies in the same plane as the axis which leads
to a meridian circle helicoid (see gure 30).
Figure 30: This gure shows a helical sweep surface with a straight line as
sweep axis and a circle as sweep object.
81
7 Results
7.2.2 General sweeps
A general sweep surface (or translation surface) is created when one or more
prole curves are moved along a leading curve. Figure 31 shows a general sweep
consisting of three straight lines forming a triangle that have been translated
along a fourth straight line.
Figure 31: This gure shows a general sweep surface with a straight line as
leading curve and three further straight lines as prole curves.
82
8 Conclusion
8 Conclusion
The aim of this thesis was to describe the newly implemented functions for
Construct3D, a 3D dynamic geometry construction tool based on the Aug-
mented Reality System Studierstube.
In the preceding chapters some software packages for static and dynamic
geometry have been presented. Afterwards the technological foundations of
this work have been described, the Open Inventor implementation Coin3D,
the Studierstube framework, Construct3D and ACIS, a geometric modeler.
Following the basic geometry principles for understanding the new curvature
toolsthe Frenet frame, the plane, the center and the circle of curvature and
the Meusnier pointas well as the helical and general sweep objects have been
explained.
Adjacent the foundations of the practial work, the design and the implemen-
tation, have been described.
Finally some gures generated by Construct3D using the new functions have
been shown.
On the one hand the author hopes that in the future students will get to
know Construct3D and use its possibilities. On the other hand, hopefully,
future students will help to expand the features of Construct3D. Anyway, may
these people gain new insights, because that is what life is for.
83
List of References
List of References
[1] Gerd Baron and Peter Kirschenhofer. Einführung in die Mathematik für
Informatiker, volume 2. Springer-Verlag, Wien, 1996.
[2] Heinrich Brauner. Lehrbuch der konstruktiven Geometrie.
Springer-Verlag, Wien, 1986.
[3] Cabrilog. Cabrilog > Products > Cabri 3D. http://www.cabri.com. last
checked: August 21, 2008.
[4] Jonathan Corney and Theodore Lim. 3D modeling with ACIS.
Saxe-Coburg Publications, Stirling, UK, 2001.
[5] Mathis Csisinko. Long Distance Distribution of Virtual and Augmented
Reality Applications . Master's thesis, Vienna University of Technology,
2006.
[6] Andreas Dünser, Hannes Kaufmann, Karin Steigbügel, and Judith
Glück. Virtual and Augmented Reality as Spatial Ability Training Tools.
In Proceedings of the CHINZ 2006, University of Canterbury,
Christchurch, New Zealand, pages 125132, 2006.
[7] Hans-Jürgen Elschenbroich. Unterrichtsentwicklung und Medieneinsatz
im Fach Mathematik: Auf dem Weg zum Lernmittelkonzept  Eine
Beratungshilfe. In Medienberatung NRW, 2006.
[8] Klaus Engel, Markus Hadwiger, Joe M. Kniss, Christof Rezk-Salama,
and Daniel Weiskopf. Real-Time Volume Graphics. A K Peters, Ltd.,
Wellesley, Massachusetts, US, 2006.
[9] Filou Software GmbH. Rhinoceros 3D - der Nurbs-Modeller unter
Windows. http://www.rhino3d.de. last checked: August 21, 2008.
[10] Georg Glaeser. Geometrie und ihre Anwendungen in Kunst, Natur und
Technik. Spektrum Akademischer Verlag, München, 2005.
[11] Andreas Goebel. Archimedes Geo3D. http://www.raumgeometrie.de.
last checked: August 21, 2008.
[12] Andreas Goebel. Archimedes Geo3D - WissensSchule.de - SHOP.
http://www.wissensschule.de/schulshop/5/54/ArchimedesGeo3D.php.
last checked: August 21, 2008.
[13] Markus Hohenwarter. GeoGebra. http://www.geogebra.org. last
checked: August 21, 2008.
[14] Hannes Kaufmann. Geometry Education with Augmented Reality. PhD
thesis, Vienna University of Technology, 2004.
[15] Benno Klotzek. Einführung in die Dierentialgeometrie. Verlag Harri
Deutsch, Frankfurt am Main, 1995.
84
List of References
[16] Ulrich H. Kortenkamp. Cinderella : Die interaktive Geometrie-Software
Cinderella. http://cinderella.de. last checked: August 21, 2008.
[17] Ulrich H. Kortenkamp and Jürgen Richter-Gebert. The interactive
geometry software Cinderella: interactive geometry on computers.
Springer-Verlag, Berlin Heidelberg, 1999.
[18] Valérie Maquil. Automatic Generation of Graphical User Interfaces in
Studierstube. Bachelor's thesis, Vienna University of Technology, 2004.
[19] McNeel. Modeling tools for designers. http://www.rhino3d.com. last
checked: August 21, 2008.
[20] Roland Mechling. Euklid DynaGeo Homepage. http://www.dynageo.de.
last checked: August 21, 2008.
[21] Parametric Technology Corporation. PTC: Pro/ENGINEER.
http://www.ptc.com. last checked: August 21, 2008.
[22] William H. Press, William T. Vetterling, Saul A. Teukolsky, and
Brian P. Flannery. Numerical Recipes in C++. The Art of Scientic
Computing. Second Edition. Cambridge University Press, 1992.
[23] Gerhard Reitmayr. On Software Design for Augmented Reality. PhD
thesis, Vienna University of Technology, 2004.
[24] Lewis F. Richardson and J. Arthur Gaunt. The Deferred Approach to
the Limit . In Philosophical Transactions of the Royal Society of London.
Series A, containing papers of a mathematical or physical character,
volume 226, pages 299361, 1927.
[25] Jürgen Richter-Gebert. Cinderella Documentation.
http://doc.cinderella.de. last checked: August 21, 2008.
[26] Dieter Schmalstieg, Anton Fuhrmann, Gerd Hesina, Zsolt Szalavari,
L. Miguel Encarnacao, Michael Gervautz, and Werner Purgathofer. The
Studierstube Augmented Reality Project . In Presence  Teleoperators and
Virtual Environments, MIT Press, volume 11, pages 3354, 2002.
[27] Sophie and Pierre René de Cotret. Cabri 3D User Manual. Cabrilog
SAS, 2005.
[28] Spatial Corporation. Spatial: 3D ACIS Modeler.
http://www.spatial.com. last checked: August 21, 2008.
[29] Karl Strubecker. Dierentialgeometrie. Kurventheorie der Ebene und des
Raumes. Walter de Gruyter & Co., Berlin, 1964.
[30] Systems in Motion AS. 3D Graphics Development Tools 
www.coin3d.org. http://www.coin3d.org. last checked: August 21, 2008.
[31] Universität Bayreuth. GEONExT: Startseite.
http://geonext.uni-bayreuth.de. last checked: August 21, 2008.
85
List of References
[32] Josie Wernecke. The Inventor Mentor  Programming Object-Oriented
3D Graphics with Open Inventor, Release 2. Addison-Wesley Longman
Publishing Co., Inc., Boston, Massachusetts, USA, 1994.
[33] Josie Wernecke. The Inventor Toolmaker  Extending Open Inventor,
Release 2. Addison-Wesley Longman Publishing Co., Inc., Boston,
Massachusetts, USA, 1994.
[34] Walter Wunderlich. Darstellende Geometrie II. Bibliographisches
Institut AG, Mannheim, 1967.
86