Dr. Scott Schaefer

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Least Squares Curves, Rational Representations,
Splines and Continuity

• Dr. Scott Schaefer

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Degree Reduction

• Given a set of coefficients for a Bezier curve
  of degree n1, find the best set of coefficients
  of a Bezier curve of degree n that
  approximate that curve

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Degree Reduction
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Degree Reduction
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Degree Reduction
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Degree Reduction
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Degree Reduction
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Degree Reduction

• Problem end-points are not interpolated

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Least Squares Optimization
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Least Squares Optimization
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Least Squares Optimization
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Least Squares Optimization
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Least Squares Optimization
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Least Squares Optimization
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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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The PseudoInverse

• What happens when isnt invertible?

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Constrained Least Squares Optimization
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Constrained Least Squares Optimization
Solution
Constraint Space
Error Function F(x)
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Constrained Least Squares Optimization
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Constrained Least Squares Optimization
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Constrained Least Squares Optimization
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Constrained Least Squares Optimization
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Constrained Least Squares Optimization
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Least Squares Curves
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Least Squares Curves
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Least Squares Curves
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Least Squares Curves
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Degree Reduction

• Problem end-points are not interpolated

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Degree Reduction
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Degree Reduction
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Rational Curves

• Curves defined in a higher dimensional space that
  are projected down

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Rational Curves

• Curves defined in a higher dimensional space that
  are projected down

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Rational Curves

• Curves defined in a higher dimensional space that
  are projected down

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Rational Curves

• Curves defined in a higher dimensional space that
  are projected down

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Why Rational Curves?

• Conics

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Why Rational Curves?

• Conics

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Why Rational Curves?

• Conics

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Why Rational Curves?

• Conics

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Derivatives of Rational Curves
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Derivatives of Rational Curves
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Derivatives of Rational Curves
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Derivatives of Rational Curves
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Splines and Continuity

• Ck continuity

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Splines and Continuity

• Ck continuity

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Splines and Continuity

• Ck continuity

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Splines and Continuity

• Ck continuity

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Splines and Continuity

• Ck continuity

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Splines and Continuity

• Assume two Bezier curves with control points
  p0,,pn and q0,,qm

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Splines and Continuity

• Assume two Bezier curves with control points
  p0,,pn and q0,,qm
• C0 pnq0

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Splines and Continuity

• Assume two Bezier curves with control points
  p0,,pn and q0,,qm
• C0 pnq0
• C1 n(pn-pn-1)m(q1-q0)

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Splines and Continuity

• Assume two Bezier curves with control points
  p0,,pn and q0,,qm
• C0 pnq0
• C1 n(pn-pn-1)m(q1-q0)
• C2 n(n-1)(pn-2pn-1pn-2)m(m-1)(q0-2q1q2)

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Splines and Continuity

• Geometric Continuity
• A curve is Gk if there exists a reparametrization
  such that the curve is Ck

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Splines and Continuity

• Geometric Continuity
• A curve is Gk if there exists a reparametrization
  such that the curve is Ck

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Splines and Continuity

• Geometric Continuity
• A curve is Gk if there exists a reparametrization
  such that the curve is Ck

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Problems with Bezier Curves

• More control points means higher degree
• Moving one control point affects the entire curve

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Problems with Bezier Curves

• More control points means higher degree
• Moving one control point affects the entire curve

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Problems with Bezier Curves

• More control points means higher degree
• Moving one control point affects the entire curve

Solution Use lots of Bezier curves and maintain
Ck continuity!!!
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Problems with Bezier Curves

• More control points means higher degree
• Moving one control point affects the entire curve

Solution Use lots of Bezier curves and maintain
Ck continuity!!!
Difficult to keep track of all the constraints. ?
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B-spline Curves

• Not a single polynomial, but lots of polynomials
  that meet together smoothly
• Local control

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B-spline Curves

• Not a single polynomial, but lots of polynomials
  that meet together smoothly
• Local control