Computer Science Department Technion-Israel Institute of Technology Introduction

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 q C(p)={x(p), y(p)}, C(0. 1) p [0, 1] C(0. 2) C(0. 7) C(0) y C(0. 4) C(0. 95) C(0. 8) C(0. 9) x C p =tangent

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

Linear Transformations Affine: Euclidean:

Linear Transformations Equi-Affine:

Differential Signatures q Euclidean invariant signature

Differential Signatures q Euclidean invariant signature

Differential Signatures q Euclidean invariant signature Cartan Theorem

Differential Signatures

~Affine

~Affine

Image transformation q Affine: q Equi-affine:

Invariant arclength should be 1. Re-parameterization invariant 2. Invariant under the group of transformations Transform Geometric measure

Euclidean arclength q Length is preserved, thus ,

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

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

Equi-affine curvature is the affine invariant curvature

Differential Signatures q Equi-affine invariant signature

From curves to surfaces q Its all about invariant measures…

Surfaces q Topology (Klein Bottle)

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

Example: Surface as graph of function q A surface,

Curves on Surfaces: The Geodesic Curvature

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

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