1. Fundamentals of Differential Geometry ( Part 2 )

2. What do the fundamental forms mean ?
Length, angle, surface area
curvatures ( deviation between the surface and the tangent plane )

3. Literature
Manfredo P. do Carmo : Differentialgeometrie von Kurven und Flächen. Vieweg, 1998

4. Curves on surfaces

5. Curves on surfaces e.g. cylinder

6. Curves on surfaces e.g. cylinder

7. tangent vector of curves on surfaces

8. Arc length of the curves on surfaces
Arc length mean the length of a parametric curve between two points defined by its parameter values t=a and t=b

9. first fundamental form

10. first fundamental form
I determines the arc length of a curve on the surface

11. first fundamental form
arc length
Angle of parametric lines
surface area

12. Length of curves on the cylinder
1. Calculation of the coefficients

13. Length of curves on the cylinder
2. Calculation of the arc length according to the curve definition

14. First fundamental form of the sphere

15. Length of curves on the sphere

16. Surface area of the sphere

17. The curvature vector of the curves on surfaces

18. The curvature vector of the curves on surfaces

19. The curvature vector of the curves on surfaces

20. Second fundamental form
II measures how far the surface is from being a plane

21. Second fundamental form
Alternative notation for the coefficients :

22. Second fundamental form of the sphere
1. Compute the normal vector

23. Second fundamental form of the sphere
2. Compute the coefficients

24. Normal curvature of surfaces
Note :
Cut the surface with the plane spanned by the tangent vector and the normal vector
->the curvature of this curve equals the normal curvature of the surface

25. Normal curvature of surfaces

26. Normal curvature of surfaces

27. Normal curvature of the sphere

28. Principal curvatures of surfaces and principal directions
are the maximum and the minimum of the normal curvature ( so-called principal curvatures ).
Principal directions are the directions of a surface in which the principal curvatures occur.

29. Elliptic Points
e. g. Ellipsoid :

30. Parabolic points e. g. cylinder :
Note. : zero principal curvatures ->
planar point of the surface
( e.g. All points of the plane )

31. Hyperbolic points
e. g. Torus :

32. curvature definitions