Approximation by Quad Meshes in Laguerre Geometry

TL;DR

This paper introduces Laguerre meshes, a novel geometric framework for approximating smooth surfaces using quadrilaterals, cones, and spheres within Laguerre geometry, extending classical mesh concepts into a new geometric setting.

Contribution

It develops the theory of Laguerre meshes, including Laguerre conjugate nets and directions, and provides methods for their computation and surface approximation.

Findings

Laguerre meshes generalize planar quadrilateral meshes in Laguerre geometry.

A method for approximating smooth surfaces with Laguerre meshes is proposed.

Laguerre conjugate nets with respect to sphere congruences are introduced.

Abstract

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.