Kilian, M., Müller, C., & Tervooren, J. (2023). Smooth and Discrete Cone-Nets. Results in Mathematics, 78(3), Article 110. https://doi.org/10.1007/s00025-023-01884-9
Cone-nets are conjugate nets on a surface such that along each individual curve of one family of parameter curves there is a cone in tangential contact with the surface. The corresponding conjugate curve network is projectively invariant and is characterized by the existence of particular transformations. We study properties of that transformation theory and illustrate how several known surface classes appear within our framework. We present cone-nets in the classical smooth setting of differential geometry as well as in the context of a consistent discretization with counterparts to all relevant statements and notions of the smooth setting. We direct special emphasis towards smooth and discrete tractrix surfaces which are characterized as principal cone-nets with constant geodesic curvature along one family of parameter curves.
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Project title:
Diskrete Flächen mit vorgeschriebener mittlerer Krümmung: P 29981-N35 (FWF Fonds zur Förderung der wissenschaftlichen Forschung (FWF)) Diskrete geometrische Strukturen motiviert durch Anwendungen: I 4868-N (FWF Fonds zur Förderung der wissenschaftlichen Forschung (FWF)) Advanced Computational Design: F 77 (FWF Fonds zur Förderung der wissenschaftlichen Forschung (FWF))
