We extend the definition of discrete conformal equivalence from triangles [5,28] to cyclic polyhedral surfaces in a straightforward way (Definition 2.2). A Euclidean cyclic polyhedral surface(, )euclidean and a hyperbolic cyclic polyhedral surface(,)˜hypar discretely conformal equivalent if. To deal with the Riemann surfaces given in terms of the Schottky data (Sec.8.2) we will need to reconstruct a function:E→R>0 satisfying (9) from the cross length ratios. It is not required that the function satisfy the triangle inequalities.) To this end, we define the auxiliary quantities associated with the angles of the triangle.
Proof This follows immediately from Theorem 2.8: The length multiratio of a quadrilateral is the modulus of the complex cross ratio.
3 Variational Principles for Discrete Conformal Maps 3.1 Discrete Conformal Mapping Problems
- Analytic Formulation of the Mapping Problems
The sides of a cyclic polygon determine its angles, but practical explicit equations for the angles as functions of the sides exist only for triangles, e.g. (21).
Given
- Variational Principles
- The Triangle Functions
In the following proposition we collect explicit formulas for the second derivatives of the functions Eg˜. Since fg is continuous and fg extends continuously to R3, the claim follows. a) First, assume that λ is contained in the feasible region of fg. Since β1, β2, β3 are constant on each connected component of the complement of the feasible region, and since.
On each component of the complement of its possible set, the function fgi is linear so that the second derivative vanishes.
4 Conformal Maps of Cyclic Quadrangulations
Emerging Circle Patterns and a Necessary Condition
Then, in the image quadrilateral, the edges incident with a black vertex meet at right angles, and the edges incident with a white vertex have the same length. One can therefore draw a circle around each white vertex through the neighboring black vertices as shown in Fig.6 (top right). Given such a circle pattern with orthogonal intersecting circles, the quadrilateral formed by edges between circle centers and intersections consists of quadrilaterals that are rectangular kites.
Therefore, the quadrilateral arising from an orthogonal circle pattern is discrete conformally equivalent (in our sense) to a combinatorially equivalent quadrilateral composed of squares. The conformal map shown in the top row of Figure 6 'finds' the orthogonal circle pattern because that circle pattern exists and the conformal map is unique (according to Theorem 3.9). For the 6×5 example in the bottom row, no corresponding orthogonal circle pattern exists.
If we map an m×nsquare grid to a parallelogram as in Figure 6, an orthogonal circle pattern will appear if mann is even. Such a pattern will not appear if one of the numbers is even and the other is odd. The increasing-angle corners and the decreasing-angle corners would have different colors.
Since the 1-skeleton is bipartite, we can assume that the vertices are colored black and white. Proof Since there are two solutions of the cross relation system with the same cross relations (see Section.2.8), there exists by Proposition2.13a functionw:V→C such that (16) holds for all edgesij∈E.
Riemann Maps with Cyclic Quadrilaterals
A boundary consists of a straight line segment containing all boundary edges of those that were also boundary edges of, and two or more lines. 8 Here we show the face circles of the solution of the Riemann map problem of Fig.7. But the face circles intersect only approximately but not exactly at right angles. line segments, each consisting of two edges that collided with a removed quadrilateral.
Each faceijmk∈incident mekis cyclic because the three vertices,j, and mare are contained in a line before the transformation. Proposition 4.2 The result of this procedure is a planar cyclic polyhedral surface that is discretely conformal equivalent to (, )eucand has its boundary polygon inscribed in a circle. The proof that the bounding polygon is inscribed in a circle is clear by construction.
Using the Möbius invariance of discrete conformal equivalence (Proposition 2.5), it is not difficult to see that the surfaces without square incident faces are discrete conformally equivalent. To show that the entire surfaces are equivalent, it suffices to show that corresponding quadrilaterals coinciding with have the same complex cross ratio. After step (2), the length cross ratio of a quadrilateral incident withki is equal to the simple length ratio between the two edges that are not incident withk.
After step (4), the cross length ratio of these edges is unchanged due to the fixed logarithmic scale factors u=0 on the neighbors ofk. After applying the Möbius transformation in step (5), the image of the point at infinity and the other three vertices of our quadrilateral event again form a cyclic square with the same complex cross ratio as at the beginning.
5 Multiply Connected Domains 5.1 Circle Domains
Special Slit Domains
The surface will only close up in the plane if the vertices to be mapped to the end points of the slot are chosen exactly right. Sometimes the symmetry of the problem determines the correct positions of the end vertices so that discrete conformal maps to slot surfaces can be computed. Patterns of horizontal lines visualize the flow of an incompressible, inviscid fluid around the hole in a channel with periodic boundary conditions.
One vertex of the triangle and the midpoint of the opposite side are mapped to the endpoints of the slit. Middle row An arrow-shaped slit is mapped to a straight slit. The two vertices at the arrow's tip, on either side of the slit, are mapped to the endpoints of the straight slit. Bottom row Three circular boundary components are mapped to horizontal slits (The slit surface is not shown.). Here we use the following trick: We start with the slit surface and map it to a surface with circular holes as described in Sect.5.1.
6 Uniformization of Spheres
Uniformizing Quadrangulations of the Sphere
11 Discrete conformal map from the cube to the sphere, calculated with the method described in Sect.6.1. We apply Möbius normalization (Sect.6.3) to the polyhedral surface with vertices on the sphere to achieve rotational symmetry.
Using the Spherical Functional
Figure 1 (above) and Figure 12 show examples of discrete conformal images of polyhedral surfaces inscribed in a sphere computed by this method.
Möbius Normalization
7 Uniformization of Tori
- Immersed Tori
- Elliptic Curves
- Choosing Points on the Sphere
- Numerical Experiments
- Putting a Square Pattern on a Spherical Mesh
The quotient space of the triangulated torus modulo the elliptic involution is then a triangulated sphere. Because the regular triangulation of the torus on the left is symmetric with respect to the elliptical involution, its image projects to a triangulation of the sphere seen on the right. Again, the triangulation on the left is symmetric with respect to the elliptical involution, so the image on the right projects to a triangulation of the sphere.
We normalize τˆ and the τˆ value obtained from the discrete uniformization so that they lie in the standard fundamental domain of the modular set, |τ|>1 and|Re(τ)|< 12, and consider the error|τ − ˆτ|. We stay away from the boundary of the fundamental domain.). In the second experiment, we choose additional points randomly to analyze how the quality of the triangulation affects the approximation error. To estimate the asymptotic behavior of the error, we determine the slope α≈ −0.88 of a line through the last four points by linear regression. Right The result of discrete uniformization after two steps of subdivision.
19 Left log-log plot of the error |τ− ˆτ| against the number of vertices for a sample of optimized random triangulations with no quality constraint. RightOnly triangulations with maxe{Qlmr(e)<0.3 are considered. We can use a variant of the discrete uniformization of elliptic curves (Section 7.2) to place a square pattern on a surface homeomorphic to a sphere. Pick four vertices of the mesh as branch points and create a bifurcated cover with two sheets of the mesh by gluing two copies along paths connecting the selected vertices.
20 "Berlin Buddy Bear" discrete, a mascot of SFB/Transregio 109 "Discretization in Geometry and Dynamics". The underlying cluster domains created by rotations around the branch points are shown on the right.
8 Uniformization of Surfaces of Higher Genus
- Fundamental Polygons and Group Generators
- From Schottky to Fuchsian Uniformization
- Hyperelliptic Curves
- Geometric Characterization of Hyperelliptic Surfaces
- Example: Deforming a Hyperelliptic Surface
- Example: Different Forms of the Same Genus-2 Surface
If all vertices of the base polygon are identified (ie, they belong to the same orbit G), then the base polygon has 4 edges. The new vertices of the underlying domain are calculated as p1=ABp0, p2=Ap0,p3=Bp0 and p4=B Ap0. The shaded region in the right image corresponds to the fundamental domain of the Schottky array in the left image.
Figure 26 shows an example of the Fuchsian uniformization of a surface of genus three represented by its Schottky uniformization. For our purposes, a hyperelliptic curve is just a bifurcated covering of the sphere λ with branch points λ1,. Triangulation of surfaces is a regular 1 to 4 subdivision of the convex hull of branch points.
A Riemann surface Rof genus ≥2 is called hyperelliptic if one of the following equivalent conditions is true (and therefore all are):. A surface is called elliptic-hyperelliptic if it is conformally equivalent to a two-sheet branched cover of the torus. Each of the holomorphic involutions has eight fixed points covering the midpoints of a pair of opposite edges.
This Lawson surface model realizes the hyperelliptic involution as a rotational Euclidean symmetry. These fixed points of the hyperelliptic involution correspond to the branching points of the hyperelliptic curve representation. We calculate a uniformization using the triangle with vertices added to the centers of the squares as shown.
The black, gray and white vertices correspond to the north and south poles of the hyperelliptic representation.
Acknowledgments This research was supported by DFG SFB/TRR 109 “Discretization in Geometry and Dynamics”. This chapter is distributed under the terms of the Creative Commons Attribution-Ncommercial 2.5 License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, distribution, and reproduction in any medium, provided that that the original author(s) and source are cited. Images or other third-party material in this chapter are licensed under a Creative Commons Work License unless otherwise noted in the credit line; if such material is not covered by a Creative Commons license for the work and such action is not permitted by law, users will need to obtain permission from the licensee to duplicate, adapt or reproduce the material.
Sechelmann, S., Bobenko, A.I., Springborn, B.: DGD Gallery, Lawson’s Surface Uniformization.https://gallery.discretization.de/models/lawsons_surface_uniformization(2015) 40.
