Differential Geometry of Surfaces

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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
Point p on the curve at u0
p
C(u)
pC(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 ? at u0 and the radius ? 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
Sv
b
?
p
Su
a
Construct an Orthonormal Basis
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Change of Coordinates
Sv
b
?
p
Su
a
First Fundamental Form
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Change of Coordinates
Sv
b
T
u
s
t
?
v
p
Su
a
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
Sv
?T is a function of direction T
b
?
p
Su
a
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Curvature
Sv
b
How do we analyze the ?T function?
?
p
Su
a
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Curvature
Eigen analysis of IIS
Eigenvalues ?1,?2
Eigenvectors E1,E2
Eigendecompostion of IIS
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Curvature
a
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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
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Weingarten Operator
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Weingarten Operator
If ?1? ?2
else umbilic (?1 ?2), 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
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Conclusion

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