1. Differential Geometry of Surfaces
Jordan Smith
UC Berkeley
CS284

2. Outline Differential Geometry of a Curve
Differential Geometry of a Surface
I and II Fundamental Forms
Change of Coordinates (Tensor Calculus)
Curvature
Weingarten Operator
Bending Energy

3. Differential Geometry of a Curve

4. Differential Geometry of a Curve
Point p on the curve at u0 p
C(u)
p=C(u0)

5. Differential Geometry of a Curve
Tangent T to the curve at u0 p Cu
C(u)

6. Differential Geometry of a Curve
Normal N and Binormal B to the curve at u0 B p Cu
Cuu
C(u) N

7. Differential Geometry of a Curve
Curvature κ at u0 and the radius ρ osculating circle B p Cu
Cuu
C(u) N

8. Differential Geometry of a Curve
Curvature at u0 is the component of -NT along T
C(u0)
C(u1) T
N(u0)
C(u)
N(u1) NT

9. Computing the Curvature of a Curve

10. Computing the Curvature of a Curve

11. Computing the Curvature of a Curve

12. Computing the Curvature of a Curve

13. Computing the Curvature of a Curve

14. Computing the Curvature of a Curve

15. Outline Differential Geometry of a Curve
Differential Geometry of a Surface
I and II Fundamental Forms
Change of Coordinates (Tensor Calculus)
Curvature
Weingarten Operator
Bending Energy

16. Differential Geometry of a Surface
S(u,v)

17. Differential Geometry of a Surface
Point p on the surface at (u0,v0) p
S(u,v)

18. Differential Geometry of a Surface
Tangent Su in the u direction p Su
S(u,v)

19. Differential Geometry of a Surface
Tangent Sv in the v direction Sv p Su
S(u,v)

20. Differential Geometry of a Surface
Plane of tangents T Sv p T Su
S(u,v)

21. First Fundamental Form IS
Metric of the surface S

22. Differential Geometry of a Surface
Normal N N Sv p T Su
S(u,v)

23. Differential Geometry of a Surface
Normal section N Sv p T Su
S(u,v)

24. Differential Geometry of a Surface
Curvature N Sv p T Su
S(u,v)

25. Differential Geometry of a Surface
Curvature NT N Sv p T Su
S(u,v)

26. Second Fundamental Form IIS

27. Outline Differential Geometry of a Curve
Differential Geometry of a Surface
I and II Fundamental Forms
Change of Coordinates (Tensor Calculus)
Curvature
Weingarten Operator
Bending Energy

28. Change of Coordinates Sv p Su
Tangent Plane of S

29. Change of Coordinates Ss St Sv b θ p a Su
Construct an Orthonormal Basis

30. Change of Coordinates Ss St Sv b θ p a Su
First Fundamental Form

31. Change of Coordinates Ss St Sv b T s u t θ v p a Su
A point T expressed in (u,v) and (s,t)

32. Outline Differential Geometry of a Curve
Differential Geometry of a Surface
I and II Fundamental Forms
Change of Coordinates (Tensor Calculus)
Curvature
Weingarten Operator
Bending Energy

33. Curvature Ss St Sv
κT is a function of direction T b θ p a Su

34. Curvature Ss St Sv
How do we analyze the κT function? b θ p a Su

35. Curvature E1 E2 φ p Ss St Su Sv a b θ Eigen analysis of IIŜ
Eigenvalues = {κ1,κ2}
Eigenvectors = {E1,E2}
Eigendecompostion of IIŜ

36. Curvature E1 E2 φ p Ss St Su Sv a b θ α

37. Outline Differential Geometry of a Curve
Differential Geometry of a Surface
I and II Fundamental Forms
Change of Coordinates (Tensor Calculus)
Curvature
Weingarten Operator
Bending Energy

38. Weingarten Operator E1 E2 φ p Ss St Su Sv a b θ

39. Weingarten Operator

40. Weingarten Operator If κ1≠ κ2
else umbilic (κ1= κ2), chose orthogonal directions

41. Outline Differential Geometry of a Curve
Differential Geometry of a Surface
I and II Fundamental Forms
Change of Coordinates (Tensor Calculus)
Curvature
Weingarten Operator
Bending Energy

42. Bending Energy

43. Bending Energy
Minimizing
= Minimizing

44. Conclusion Curvature of Curves and Surfaces
Computing Surface Curvature using the Weingarten Operator
Minimizing Bending Energy
Gauss-Bonnet Theorem