Constrained Modelling of 3‐Valent Meshes Using a Hyperbolic Deformation Metric

Abstract

Polygon meshes with 3‐valent vertices often occur as the frame of free‐form surfaces in architecture, in which rigid beams are connected in rigid joints. For modelling such meshes, it is desirable to measure the deformation of the joints' shapes. We show that it is natural to represent joint shapes as points in hyperbolic 3‐space. This endows the space of joint shapes with a geometric structure that facilitates computation. We use this structure to optimize meshes towards different constraints, and we believe that it will be useful for other applications as well.Polygon meshes with 3‐valent vertices often occur as the frame of free‐form surfaces in architecture, in which rigid beams are connected in rigid joints.