Introduction to Differential Geometry Computer Science Department Technion-Israel Institute of Technology Introduction to Differential Geometry Ron Kimmel www.cs.technion.ac.il/~ron Geometric Image Processing Lab

Planar Curves C(p)={x(p),y(p)}, p [0,1] C(0.1) C(0.2) C =tangent

Arc-length and Curvature s(p)= | |dp C

Linear Transformations Affine: Euclidean: Euclidean Affine Affine

Linear Transformations Equi-Affine: Euclidean Equi-Affine Equi-Affine

Differential Signatures Euclidean invariant signature Euclidean

Differential Signatures Euclidean invariant signature Euclidean

Differential Signatures Euclidean invariant signature Euclidean Cartan Theorem

Differential Signatures

~Affine

~Affine

Image transformation Affine: Equi-affine:

Invariant arclength should be Re-parameterization invariant Invariant under the group of transformations Geometric measure Transform

Euclidean arclength Length is preserved, thus ,

Euclidean arclength Length is preserved, thus re-parameterization invariance Length is preserved, thus

Equi-affine arclength re-parameterization invariance Area is preserved, thus

Equi-affine curvature is the affine invariant curvature

Differential Signatures Equi-affine invariant signature Equi-Affine

From curves to surfaces Its all about invariant measures…

Surfaces Topology (Klein Bottle)

Surface A surface, For example, in 3D Normal Area element Total area

Example: Surface as graph of function A surface,

Curves on Surfaces: The Geodesic Curvature

Curves on Surfaces: The Geodesic Curvature Normal Curvature Principle Curvatures Gauss Mean Curvature Gaussian Curvature

Geometric measures www.cs.technion.ac.il/~ron Curvature k, normal , tangent , arc-length s Mean curvature H Gaussian curvature K principle curvatures geodesic curvature normal curvature tangent plane www.cs.technion.ac.il/~ron