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Differential Geometry of Surfaces Jordan Smith UC Berkeley CS284.

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Presentation on theme: "Differential Geometry of Surfaces Jordan Smith UC Berkeley CS284."— Presentation transcript:

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 C(u)

4 Differential Geometry of a Curve p C(u) p=C(u 0 ) Point p on the curve at u 0

5 Differential Geometry of a Curve C(u) CuCu p Tangent T to the curve at u 0

6 Differential Geometry of a Curve C(u) CuCu C uu N p Normal N and Binormal B to the curve at u 0 B

7 Differential Geometry of a Curve C(u) CuCu C uu N p Curvature κ at u 0 and the radius ρ osculating circle B

8 Differential Geometry of a Curve C(u) T NTNT Curvature at u 0 is the component of -N T along T C(u 0 ) C(u 1 ) N(u 0 ) N(u 1 )

9 Computing the Curvature of a Curve

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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 S(u,v) p Point p on the surface at (u 0,v 0 )

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

19 Differential Geometry of a Surface S(u,v) p SuSu SvSv Tangent S v in the v direction

20 Differential Geometry of a Surface S(u,v) p Plane of tangents T SuSu SvSv T

21 First Fundamental Form I S Metric of the surface S

22 Differential Geometry of a Surface S(u,v) N p SuSu SvSv T Normal N

23 Differential Geometry of a Surface S(u,v) N p SuSu SvSv T Normal section

24 Differential Geometry of a Surface S(u,v) N p Curvature SuSu SvSv T

25 Differential Geometry of a Surface S(u,v) N p SuSu SvSv T NTNT Curvature

26 Second Fundamental Form II S

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 p SuSu SvSv Tangent Plane of S

29 Change of Coordinates p SsSs StSt SuSu SvSv a b θ Construct an Orthonormal Basis

30 Change of Coordinates p SsSs StSt SuSu SvSv a b θ First Fundamental Form

31 Change of Coordinates p SsSs StSt SuSu SvSv a b θv u t s T 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 p SsSs StSt SuSu SvSv a b θ κ T is a function of direction T

34 Curvature p SsSs StSt SuSu SvSv a b θ How do we analyze the κ T function?

35 Curvature E1E1 E2E2 φ p SsSs StSt SuSu SvSv a b θ Eigen analysis of II Ŝ Eigenvalues = {κ 1,κ 2 } Eigenvectors = {E 1,E 2 } Eigendecompostion of II Ŝ

36 Curvature E1E1 E2E2 φ p SsSs StSt SuSu SvSv 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 E1E1 E2E2 φ p SsSs StSt SuSu SvSv a b θ

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40 If κ 1 ≠ κ 2 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 Minimizing= Minimizing

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


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