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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**

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**Differential Geometry of a Curve**

Point p on the curve at u0 p C(u) p=C(u0)

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**Differential Geometry of a Curve**

Tangent T to the curve at u0 p Cu C(u)

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**Differential Geometry of a Curve**

Normal N and Binormal B to the curve at u0 B p Cu Cuu C(u) N

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**Differential Geometry of a Curve**

Curvature κ at u0 and the radius ρ osculating circle B p Cu Cuu C(u) N

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**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

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**Computing the Curvature of a Curve**

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**Computing the Curvature of a Curve**

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**Computing the Curvature of a Curve**

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**Computing the Curvature of a Curve**

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**Computing the Curvature of a Curve**

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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**

Point p on the surface at (u0,v0) p S(u,v)

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**Differential Geometry of a Surface**

Tangent Su in the u direction p Su S(u,v)

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**Differential Geometry of a Surface**

Tangent Sv in the v direction Sv p Su S(u,v)

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**Differential Geometry of a Surface**

Plane of tangents T Sv p T Su S(u,v)

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**First Fundamental Form IS**

Metric of the surface S

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**Differential Geometry of a Surface**

Normal N N Sv p T Su S(u,v)

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**Differential Geometry of a Surface**

Normal section N Sv p T Su S(u,v)

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**Differential Geometry of a Surface**

Curvature N Sv p T Su S(u,v)

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**Differential Geometry of a Surface**

Curvature NT N Sv p T Su S(u,v)

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**Second Fundamental Form IIS**

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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 Sv p Su Tangent Plane of S

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**Change of Coordinates Ss St Sv b θ p a Su**

Construct an Orthonormal Basis

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

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**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)

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

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

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**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

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

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Weingarten Operator

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**Weingarten Operator If κ1≠ κ2**

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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Bending Energy 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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