Advanced Topics in Computer Graphics: Discrete Differential Geometry (600.657)

Course Description

In this course, we will focus on the challenges of applying differential geometric techniques in the discrete setting. In particular, we will look at methods for defining discrete analogues of differentiable constructs that are guaranteed to preserve important invariants directly in the discrete setting, and not just in the limit of the discretization process. Throughout the semester, we will consider several application domains including surface evolution, physical modeling, conformal geometry, and meshing.

Notes

The slides for this course borrow heavily from Mathieu Desbrun's course notes at Caltech and from the discrete differential geometry courses at SIGGRAPH (2006) and SIGGRAPH Asia (2008). Any errors in these notes are likely my own introduction and my adaptation of these courses should be taken with a grain of salt.

Date	Subject	Notes
9/02/09	Introduction	Notes
9/09/09	Curves	Notes
9/14/09	Surfaces	Notes
9/16/09	Surfaces	Notes
9/21/09	Surfaces	Notes
9/23/09	Mean Curvature Flow	Notes
9/28/09	Class Cancelled (Yom Kippur)
9/30/09	Harmonic Maps	Notes
10/05/09	Intrinsic Laplacian Operators	Notes
10/07/09	Discrete Thin Shells	Notes
10/12/09	Conformal Maps	Notes
10/14/09	Circle Packings	Notes
10/19/09	Circle Patterns	Notes
10/21/09	Circle Patterns	Notes
10/26/09	Class Cancelled
10/28/09	Integration and Surface Flow	Notes
11/02/09	Willmore Flow	Notes
11/04/09	Willmore Flow	Notes
11/09/09	Class Cancelled
11/11/09	Project Proposal Presentations
11/16/09	Conformal Metrics	Notes
11/18/09	Discrete Exterior Calculus	Notes
11/23/09	Discrete Exterior Calculus	Notes
11/30/09	Stable Fluids	Notes
12/02/09	Discrete One-Forms and Tutte's Embedding

Reading

Convex Representations of Graphs
How to Draw a Graph
On the Curvature of Piecewise Flat Spaces
An Algorithm for Evolutionary Surfaces
Computing Discrete Minimal Surfaces and Their Conjugates
Multiresolution Analysis of Arbitrary Meshes
On the Convergence of Circle Packings to the Riemann Map
Parametrization and Smooth Approximation of Surface Triangulations
The Approximation of Conformal Structures via Circle Packing
Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow
Discrete Differential-Geometry Operators for Triangulated 2-Manifolds
Least Squares Conformal Maps for Automatic Texture Atlas Generation
Intrinsic Parameterizations of Surface Meshes
A Circle Packing Algorithm
Discrete Exterior Calculus
Restricted Delaunay Triangulations and Normal Cycle
Variational Principles for Circle Patterns and Koebe's Theorem
Discrete Differential Forms for Computational Modeling
Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization
Discrete Willmore Flow
A Discrete Laplace-Beltrami Operator for Simplicial Surfaces
Designing Quadrangulations with Discrete Harmonic Forms
On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces
Geometric Modeling with Conical Meshes and Developable Surfaces
Discrete Conformal Mappings via Circle Patterns
Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization
Discrete Laplace operators: No free lunch
An Algorithm for the Construction of Intrinsic Delaunay Triangulations with Applications to Digital Geometry Processing
Stable, Circulation-Preserving, Simplicial Fluids
Discrete Elastic Rods
Discrete Laplace Operator on Meshed Surfaces
Conformal Flattening by Curvature Prescription and Metric Scaling
SIGGRAPH Asia Course Notes (2008)
A Curvature Theory for Discrete Surfaces Based on Mesh Parallelity