Approximation by Quad Meshes in Laguerre Geometry
This work addresses surface approximation in computational geometry, specifically for applications in design and modeling, but appears incremental as it extends existing concepts to Laguerre geometry.
The paper tackles the problem of approximating smooth surfaces using Laguerre meshes, which are watertight surfaces composed of planar quadrilaterals, cone strips, and spherical faces, by introducing Laguerre conjugate directions and a computational method, resulting in an analog of conjugate nets in Laguerre geometry.
We study analogs of planar-quadrilateral meshes in Laguerre sphere geometry and the approximation of smooth surfaces by them. These new Laguerre meshes can be viewed as watertight surfaces formed by planar quadrilaterals (corresponding to the vertices of a mesh), strips of right circular cones (representing the edges), and spherical faces. In the smooth limit, we get an analog of conjugate nets in Laguerre geometry, which we call Laguerre conjugate nets with respect to an attached sphere congruence. We introduce the notion of Laguerre conjugate directions, provide a method for computing them, and apply them to approximate surfaces by L-meshes with prescribed radii of spherical faces.