Differential Geometry Curve

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Differential Geometry (Curve)
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Definitions
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Smooth Curve
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Velocity, Speed
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Arc Length
or
Unit-speed curve has arc-length parameterization
a
t
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Reparameterization
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Arc-length Reparameterization
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Arc Length Parameterization
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Tangent, Normal (plane curve)
T
N
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t 1
t 0
t 1
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Curvature
slope angle f
(/0/)
curvature
absolute curvature
We knew
Therefore,
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Ex Curvature
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Curvature of Arbitrary-Speed Curves
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Summary (Planar Curve)

• Unit-speed (arc-length parameterization)

• Arbitrary-speed

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Space Curve
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Frenet Frame (Farin)
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Frenet-Serret Formula
Express TNB in terms of TNB
?
?
?
Orthonormal expansion
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Geometric Meaning of k and t
x(sDs)
Da angle between t(s) and t(sDs)
Db angle between b(s) and b(sDs)
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Euclidean Invariant
Not changed by a Euclidean transformation (rigid
body transform)
Any two continuous function k(s) and t(s)
uniquely define a curve
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Approximation of Space Curves
the linear approximation of b
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Arbitrary-Speed Curve
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Ex Cubic Bezier Curve (ref)
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Rotation-Minimizing Frame

• Applications sweep, axial deformation

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• Klok (86)

• ???(98)

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Algebraic Curve

• Parameterization
• Implicitization

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