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Differential Geometry of Surfaces Jordan Smith UC Berkeley CS284

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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

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Differential Geometry of a Curve C(u)

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

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

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

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

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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 )

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Computing the Curvature of a Curve

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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

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Differential Geometry of a Surface S(u,v)

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Differential Geometry of a Surface S(u,v) p Point p on the surface at (u 0,v 0 )

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

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

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

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First Fundamental Form I S Metric of the surface S

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

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

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

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

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Second Fundamental Form II S

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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

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Change of Coordinates p SuSu SvSv Tangent Plane of S

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

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

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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)

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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

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Curvature p SsSs StSt SuSu SvSv a b θ κ T is a function of direction T

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

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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

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

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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

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Weingarten Operator E1E1 E2E2 φ p SsSs StSt SuSu SvSv a b θ

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If κ 1 ≠ κ 2 else umbilic (κ1= κ2), chose orthogonal directions

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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

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Bending Energy

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Minimizing= Minimizing

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

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