1. Introduction to Differential Geometry
Computer Science Department
Technion-Israel Institute of Technology
Introduction to Differential Geometry
Ron Kimmel
Geometric Image Processing Lab

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

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

4. Linear Transformations
Affine:
Euclidean:
Euclidean
Affine
Affine

5. Linear Transformations
Equi-Affine:
Euclidean
Equi-Affine
Equi-Affine

6. Differential Signatures
Euclidean invariant signature
Euclidean

7. Differential Signatures
Euclidean invariant signature
Euclidean

8. Differential Signatures
Euclidean invariant signature
Euclidean
Cartan Theorem

9. Differential Signatures

10. ~Affine

11. ~Affine

12. Image transformation
Affine:
Equi-affine:

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

14. Euclidean arclength
Length is preserved, thus ,

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

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

17. Equi-affine curvature
is the affine invariant curvature

18. Differential Signatures
Equi-affine invariant signature
Equi-Affine

19. From curves to surfaces
Its all about invariant measures…

20. Surfaces
Topology (Klein Bottle)

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

22. Example: Surface as graph of function
A surface,

23. Curves on Surfaces: The Geodesic Curvature

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

25. 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