Introduction to Differential Geometry

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Introduction to Differential Geometry
Computer Science Department
Technion-Israel Institute of Technology

• Ron Kimmel
• www.cs.technion.ac.il/ron

Geometric Image Processing Lab
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Planar Curves

• C(p)x(p),y(p), p 0,1

C(0.1)
C(0.2)
C(0.7)
C(0)
C(0.4)
C(0.8)
C(0.95)
y
C(0.9)
x
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Arc-length and Curvature

• s(p) dp

C
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Linear Transformations
Affine
Euclidean
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Linear Transformations
Equi-Affine
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Differential Signatures

• Euclidean invariant signature

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Differential Signatures

• Euclidean invariant signature

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Differential Signatures

• Euclidean invariant signature

Cartan Theorem
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Differential Signatures
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Affine
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Affine
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Image transformation

• Affine
• Equi-affine

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Invariant arclength should be

1. Re-parameterization invariant
2. Invariant under the group of transformations

Transform
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Euclidean arclength

• Length is preserved, thus ,

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Euclidean arclength

• Length is preserved, thus

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Equi-affine arclength

• Area is preserved, thus

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Equi-affine curvature
is the affine invariant curvature
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Differential Signatures

• Equi-affine invariant signature

Equi-Affine
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From curves to surfaces

• Its all about invariant measures

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Surfaces

• Topology (Klein Bottle)

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Surface

• A surface,
• For example, in 3D
• Normal
• Area element
• Total area

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Example Surface as graph of function

• A surface,

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Curves on Surfaces The Geodesic Curvature
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Curves on Surfaces The Geodesic Curvature
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Geometric measures

• Curvature k, normal , tangent ,
  arc-length s
• Mean curvature H
• Gaussian curvature K
• principle curvatures
• geodesic curvature
• normal curvature
• tangent plane

• www.cs.technion.ac.il/ron