Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximation by Quad Meshes in Laguerre Geometry

A. Ramos-Cisneros, M. Skopenkov, H. Pottmann

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.

Approximation by Quad Meshes in Laguerre Geometry

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.
Paper Structure (46 sections, 4 theorems, 40 equations, 21 figures, 1 table)

This paper contains 46 sections, 4 theorems, 40 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions and overview
  3. Related work
  4. A few basics of Laguerre geometry and L-nets
  5. Laguerre Geometry in Euclidean 3-space
  6. Oriented spheres
  7. Oriented planes
  8. Oriented contact
  9. Laguerre transformations
  10. Oriented cones
  11. Cyclographic Model
  12. Lines in Minkowski space
  13. Laguerre transformations
  14. L-nets
  15. Examples of L-nets
  16. ...and 31 more sections

Key Result

Theorem 3.4

Two non-parallel tangent vectors $A=a_1S_u+a_2S_v$ and $B=b_1S_u+b_2S_v$ at a surface point $S(u,v)\in \mathbb{R}^{3,1}$ are L-conjugate with respect to the attached congruence $P$ if and only if where and $N(u,v)$ is a normal vector to $P(u,v)$ smoothly depending on $(u,v)$.

Figures (21)

  • Figure 1: An L-mesh represents a smooth surface in positively curved areas, and consists of planar vertex quads (blue), spherical faces bounded by circular arcs (red) and pieces of rotational cones (beige), which we view as its edges. The presence of orthogonal conical patches implies that the overall pattern is smooth for an L-mesh, which is a discrete principal parameterization (left), while in a more general case, one has a staircase pattern (right). The sphere set is fair, as illustrated by the top view of the sphere center mesh (top row), shown as a gray mesh with black edges and red vertices.
  • Figure 2: Roof concept design based on a portion of an L-net (shown in Fig. \ref{['fig:spiral_defM']}), illustrating the watertight surface generated from the corresponding L-net structure.
  • Figure 3: L-net approximation of a spiral-like surface. Smaller spherical panels appear in the narrow regions of the surface, whereas larger spherical panels arise in the wider regions.
  • Figure 4: L-net approximation of a seashell-like surface.
  • Figure 5: A Laguerre transformation of the L-net in Fig. \ref{['fig:seashellex5']}. The inset shows a close-up of a region where singularities appear. Laguerre transformations allow us to explore different designs, but singularities can appear in the process.
  • ...and 16 more figures

Theorems & Definitions (17)

  • Definition 2.1: L-net
  • Definition 3.1: L-conjugate net
  • Remark 3.2
  • Definition 3.3: L-conjugate tangents
  • Theorem 3.4
  • proof
  • Definition 3.5: Self-L-conjugate tangents
  • Definition 3.6
  • Proposition 3.7
  • proof
  • ...and 7 more