Discrete Surface Geometry and Intrinsic Triangulations

Explore the fundamentals of discrete surface geometry and intrinsic triangulations in this 59-minute Fields Postdoc Colloquium talk by Nicholas Sharp. Delve into the world of piecewise-flat surfaces and discover how intrinsic triangulations offer a unique perspective on representing 3D shapes. Learn about the bridge between geometric theory and practical algorithms in geometric computing, starting with basic concepts and progressing to advanced topics. Examine surface meshes, gradients, vector fields, and Gaussian curvature before diving into intrinsic edge lengths and their historical context. Investigate Euclidean intrinsic triangles, cone metrics, and non-embeddable triangulations. Understand key properties, including Delaunay edge flips and refinement techniques. Explore applications in nonmanifold meshes, robustness improvements, and algorithm enhancements. No prior knowledge of the subject is required for this comprehensive introduction to discrete surface geometry and its practical implications in science and engineering.

Intro
Surface meshes
Gradients and vector fields
There is no perfect mesh
Robust geometry processing
Gaussian curvature revisited
Basic idea intrinsic edge lengths
A brief history of intrinsic triangulations
Euclidean intrinsic triangles a larger space of triangulations
Intrinsic is enough
Perspective: cone metrics
A non-embeddable intrinsic triangulatio
Key idea: a larger space of triangulation
Intrinsic edge flips
Properties of intrinsic triangulations
Better intrinsic meshes
Delaunay edge flips
Proof sketch
Intrinsic Delaunay triangulations many characterizations & properties
Better basis functions
A-complex
Intrinsic Delaunay refinement
Applications
Nonmanifold intrinsic triangulations
Robustness as a subroutine build a better Laplace matrix
Nonmanifold meshes
Resolving nonmanifoldness assembling the tufted cover
The tufted cover vertex-nonmanifold almost everywhere
Improving algorithms
Properties bounded interpolation
Delaunay flipping distance a motivating example