Developable Quad Meshes and Contact Element Nets

Victor Ceballos Inza, Florian Rist, Johannes Wallner, Helmut Pottmann

Research output: Contribution to journalArticlepeer-review

Abstract

The property of a surface being developable can be expressed in different equivalent ways, by vanishing Gauss curvature, or by the existence of isometric mappings to planar domains. Computational contributions to this topic range from special parametrizations to discrete-isometric mappings. However, so far a local criterion expressing developability of general quad meshes has been lacking. In this paper, we propose a new and efficient discrete developability criterion that is applied to quad meshes equipped with vertex weights, and which is motivated by a well-known characterization in differential geometry, namely a rank-deficient second fundamental form. We assign contact elements to the faces of meshes and ruling vectors to the edges, which in combination yield a developability condition per face. Using standard optimization procedures, we are able to perform interactive design and developable lofting. The meshes we employ are combinatorially regular quad meshes with isolated singularities but are otherwise not required to follow any special curves on a developable surface. They are thus easily embedded into a design workflow involving standard operations like remeshing, trimming, and merging operations. An important feature is that we can directly derive a watertight, rational bi-quadratic spline surface from our meshes. Remarkably, it occurs as the limit of weighted Doo-Sabin subdivision, which acts in an interpolatory manner on contact elements.

Original languageEnglish (US)
Article number183
JournalACM transactions on graphics
Volume42
Issue number6
DOIs
StatePublished - Dec 4 2023

Keywords

  • checkerboard pattern
  • computer-aided design
  • contact elements
  • developable surface
  • discrete differential geometry
  • shape optimization

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design

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