The discrete differential geometry of checkerboard patterns

Abstract

Checkerboard patterns are quadrilateral nets where every second face is a parallelogram. They are easily constructed from a general quadrilateral net, the so-called control net, by a midpoint subdivision. Checkerboard patterns have already proven to be very useful in various applications in geometric modeling or architecture suggesting that there is more to the approach than just good numeric approximation properties. It is the aim of this thesis to develop a discrete differential geometric theory for checkerboard patterns. We lay a special focus on the transformation group principle saying that the discrete structures should be invariant under the same transformations as their smooth counter parts. A curvature theory is developed for checkerboard patterns based on a novel definition of the shape operator. This piecewise linear operator maps the parallelograms in a checkerboard pattern to the parallelograms in its Gauss image. We define conjugate, orthogonal and principal checkerboard patterns and find that they are consistent with the introduced shape operator. Conjugacy is preserved under projective transformations applied to the control net. Orthogonal checkerboard patterns can be identified with sphere congruences of orthogonally intersecting spheres. Möbius transformations can be applied to these spheres preserving orthogonality and the class of principal checkerboard patterns. The special case of rhombic checkerboard patterns is discussed from another perspective. Orthogonality of the control net can be expressed through a rhombic checkerboard pattern making this approach particularly interesting for applications. Ivory’s Theorem guarantees the existence of orthogonal multi-nets in R2 where the orthogonality constrained is met by any parameter rectangle. We extend this approach to R3 and show the versatile use-cases of rhombic checkerboard patterns. Koenigs checkerboard patterns are those checkerboard patterns where the six focal points of every face lie on a common conic section, the so-called conic of Koenigs. A discrete dualization can be naturally defined for checkerboard patterns by mapping the parallelograms to a scaled version of themselves with reversed orientation. We find that Koenigs checkerboard patterns are precisely the dualizable ones. Discrete analogues of the Laplace invariants can be defined on the edges of the two diagonal nets of the control net of a checkerboard pattern. It turns out that Koenigs checkerboard patterns can be characterized by the fact that the Laplace invariants defined on corresponding diagonals are equal. On the one hand this fits into the classical theory and on the other hand this shows that the class of Koenigs checkerboard patterns is invariant under projective transformations applied to the control net. Finally, isothermic checkerboard patterns are defined as principal Koenigs checkerboard patterns. We find that the class of isothermic checkerboard patterns is invariant under both dualization and Möbius transformations. This allows to generate discrete minimal surfaces from isothermic checkerboard patterns in the plane. First they are mapped to the unit sphere by a Möbius transformation and then dualization yields discrete minimal surfaces.