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Discrete Differential Geometry
Qixing Huang
March. 2th 2017
Slide Credit: Mirela Ben-Chen
Motivation
Motivation
Differential Geometry
Parametric Curves
Parametric Curves
Length of a Curve
Examples
Arc Length Parameterization
Curvature
Examples
The Normal Vector
The Osculating Plane
The Binormal Vector
The Frenet Frame
The Frenet Frame
The Frenet Frame
Curvature and Torsion
Surfaces
Differential Geometry: Surfaces
Differential Geometry: Surfaces
Normal Curvature
Surface Curvature
Principal Curvature
Curvature
Surface Classification
Principal Directions
Gauss-Bonnet Theorem
Gauss-Bonnet Theorem Example
Fundamental Forms
Fundamental Forms
Fundamental Forms
Intrinsic Geometry
• Properties of the surface that only depend on the first fundamental form
– length
– angles
– Gaussian curvature
Laplace-Beltrami Operator
Laplace-Beltrami Operator
Discrete Differential Operators
Functions on Meshes
Gradient of a Function
Gradient of a Function
Discrete Laplace-Beltrami First Approach
Discrete Laplace-BeltramiUniform Discretization
Discrete Laplace-Beltrami--Second Approach
Discrete Laplace-Beltrami Cotangent Formula
Discrete Normal
• Mean curvature
• Gaussian curvature
• Principal curvatures
Discrete Curavtures
Example: Mean Curvature
Example: Gaussian Curvature