Surface discretization & Lie spheregeometry
Wayne Rossman, Kobe Univ, Nov 15 2018
Goal:
Discretize surfaces while preserving
mathematical structuressss佐藤

Result by Teramoto , and Umehara, Murata• Front = Legendre immersion• Frontal = normal field well defined but not necessarily a Legendre immersion• Nondegenerate singularity “=“ local singular set is a single creased curve• Teramoto’s and Umehara-Murata’s result: For any nondegenerate singularity on afront, one of the principal curvatures will diverge to infinity, with a sign change.

When are discrete flat or parabolic orsingular vertices necessarily singular?
• Smooth nonzero-CGC surfaces never have a principal curvature equal to zero.
• Parallel surfaces of smooth minimal surfaces in R^3 without umbilic points never
have a principal curvature equal to zero.

• Parallel surfaces of smooth maximal surfaces in R^(2,1) without umbilic points never
have a principal curvature equal to zero.

Examples of discrete Ω surfaces• Minimal in R^3 and H^3

Examples of discrete Ω surfaces• Minimal in R^3 and H^3• Delaunay in R^3 and S^3 (including Clifford minimal)

Examples of discrete Ω surfaces• Minimal in R^3 and H^3• Delaunay in R^3 and S^3 (including Clifford minimal)• Constant Gaussian curvature, in R^3

Examples of discrete Ω surfaces• Minimal in R^3 and H^3• Delaunay in R^3 and S^3 (including Clifford minimal)• Constant Gaussian curvature, in R^3• Linear Weingarten of Bryant type in H^3

Examples of discrete Ω surfaces• Minimal in R^3 and H^3• Delaunay in R^3 and S^3 (including Clifford minimal)• Constant Gaussian curvature, in R^3• Linear Weingarten of Bryant type in H^3• Higher-order Enneper minimal surface in R^3, and parallel surface (with a 1/3 part)Examples of discrete Ω surfaces• Minimal in R^3 and H^3• Delaunay in R^3 and S^3 (including Clifford minimal)• Constant Gaussian curvature, in R^3• Linear Weingarten of Bryant type in H^3• Higher-order Enneper minimal surface in R^3, and parallel surface (with a 1/3 part)• Flat in H^3, higher-order Enneper-type maximal in R^(2,1), and parallel surface

Examples of results about singularities on
discrete Ω surfaces• Weierstrass representations for discrete surfaces in all cases where we expect them
(Bobenko, Pinkall, Hertrich-Jeromin --->Hoffmann, Sasaki, Yoshida, Yasumoto & I).

Examples of results about singularities on
discrete Ω surfaces• Weierstrass representations for discrete surfaces in all cases where we expect them.
• Smooth flat surfaces in H^3 have caustics that are also flat, with a Weierstrass
representation for the caustics. We (Hoffmann, Sasaki, Yoshida, R--) proved acorresponding setting in the discrete case. Here the caustics have a discrete
Weierstrass representation and have planar quadrilaterals, but not circular ones. In thediscrete case as well, it’s the collection of singular vertices on the parallel surfaces ofthe flat surface that form the caustic.

Examples of results about singularities ondiscrete Ω surfaces• Weierstrass representations for discrete surfaces in all cases where we expect them.
• Weierstrass representation for caustics of discrete flat surfaces in H^3 matching withsingular vertices on the parallel surfaces of the initial flat surface.

• Singularities on smooth maximal surfaces in R^(2,1) occur at lightlike points of thesurface, where g (stereographic projection of the Gauss map) has absolute value 1.
Yasumoto and I: singular vertices on discrete maximal surfaces have adjacent facesthat are not spacelike, where the corresponding quadrilateral for the discrete g has acircumcircle that intersects S^1 (the last part Yasumoto established in a solo paper).

Examples of results about singularities ondiscrete Ω surfaces• Weierstrass representations for discrete surfaces in all cases where we expect them.
• Weierstrass representation for caustics of discrete flat surfaces in H^3 matching withsingular vertices on the parallel surfaces of the initial flat surface.

• Singular vertices on discrete maximal surfaces have adjacent faces that are notspacelike, where the corresponding circumcircle for the discrete g intersects S^1.

• Similar result (Yasumoto & I) for “spacelike” CMC 1 surfaces in de Sitter spaceS^(2,1).

• Still cannot distinguish a cuspidal edge from a swallowtail in the discrete case!