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