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. 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. This structure can be used to optimize meshes towards different constraints, and it might be useful for other applications as well.
Ronald Richter, Jan Eric Kyprianidis, Boris Springborn, and Marc Alexa.
Constrained Modelling of 3-Valent Meshes Using a Hyperbolic Deformation Metric.
Computer Graphics Forum, 2016.