Parametric Curves

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Parametric Curves

• Ref 1, 2

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Outline

• Hermite curves
• Bezier curves
• Catmull-Rom splines
• Frames along the curve

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Hermite Curves

• 3D curve of polynomial bases
• Geometrically defined by position and tangents of
  end points
• Able to construct C1 composite curve
• In CG, often used as the trace for camera with
  Frenet frame, or rotation-minimizing frame

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Math

• h1(s) 2s3 - 3s2 1
• h2(s) -2s3 3s2
• h3(s) s3 - 2s2 s
• h4(s) s3 - s2
• P(s) P1h1(s) P2h2(s) T1h3(s) T2h4(s)
• P(s)P1h1(s) P2h2(s) T1h3(s) T2h4(s)
• h1 6s2-6s h2 -6s26s h3 3s2-4s1
  h4 3s2 2s

P(0) P1, P(1)P2 P(0)T1, P(1)T2
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Blending Functions
h2(s)
h4(s)
h3(s)
h1(s)

• At s 0
• h1 1, h2 h3 h4 0
• h1 h2 h4 0, h3 1
• At s 1
• h1 h3 h4 0, h2 1
• h1 h2 h3 0, h4 1

P(0) P1 P(0) T1
P(1) P2 P(1) T2
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C1 Composite Curve
P(t)
Q(t)
R(t)
More on Continuity
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Composite Curve
Each subcurve is defined in 0,1. The whole
curve (PQR) can be defined from 0,3 To
evaluate the position (and tangent)
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Close Relatives

• Bezier curves
• Catmull-Rom splines

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Bezier Curve (cubic, ref)

• Defined by four control points
• de Casteljau algorithm (engineer at Citroën)

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Bezier Curve (cont)

• Also invented by Pierre Bézier (engineer of
  Renault)
• Blending function Bernstein polynomial
• Can be of any degree

Degree n has (n1) control points
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First Derivative of Bezier Curves (ref)

• Degree-n Bezier curve
• Bernstein polynomial
• Derivative of Bernstein polynomial
• First derivative of Bezier curve
• Hodograph

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Ex cubic Bezier curve
Hence, to convert to/from Hermite curve
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C1 Composite Bezier Curves
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Bezier Curve Fitting

• From Graphics Gems (code)
• Input digitized data points in R2
• Output composite Bezier curves in specified error

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Bezier Marching

• A path made of composite Bezier curves
• Generate a sequence of points along the path with
  nearly constant step size
• Adjust the parametric increment according to
  (approximated) arc length

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Catmull-Rom spline (1974, ref)

• Given n1 control points P0,,Pn, we wish to
  find a curve that interpolates these control
  points (i.e. passes through them all), and is
  local in nature (i.e. if one of the control
  points is moved, it only affects the curve
  locally).
• We define the curve on each segment Pi,Pi1 by
  using the two control points, and specifying the
  tangent to the curve at each control point to be
  (Pi1Pi-1)/2 and (Pi2Pi)/2
• Tangents in first and last segments are defined
  differently

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PowerPoint Line Tool

• Gives you a Catmull-Rom spline, open or close.

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Ex Catmull-Rom Curves
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Reference Frames Along the Curve

• Applications
• generalized cylinder
• Cinematography
• Frenet frames
• Rotation minimizing frame

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Generalized Cylinder
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Frenet Frame (Farin)
? cross product
Unit vectors
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Frenet Frame (arc-length parameterization)
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Frenet-Serret Formula
In this notation, the curve is r(s)
Express TNB (change rate of TNB) in terms of
TNB
?
?
?
Orthonormal expansion
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Frenet-Serret Formula (cont)
In general parameterization r(t)
Curvature and torsion
r(t)(x(t),y(t))
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Geometric Meaning of k and t
curvature
torsion
x(sDs)
Da angle between t(s) and t(sDs)
Db angle between b(s) and b(sDs)
More result on this reference
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Frenet Frame Problem

• Problem vanishing second derivative at
  inflection points
• (vanishing normal)

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Rotation Minimizing Frame (ref)

• Use the second derivative to define the first
  frame (if zero, set N0 to any vector ?T0)
• Compute all subsequent Ti find a rotation from
  Ti-1 to Ti rotate Ni and Bi accordingly
• If no rotation, use the same frame

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Continuity

• Geometric Continuity
• A curve can be described as having Gn continuity,
  n being the increasing measure of smoothness.
• G0 The curves touch at the join point.
• G1 The curves also share a common tangent
  direction at the join point.
• G2 The curves also share a common center of
  curvature at the join point.

• Parametric Continuity
• C0 curves are joined
• C1 first derivatives are equal
• C2 first and second derivatives are equal
• Cn first through nth derivatives are equal

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