Dr' Scott Schaefer

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Catmull-Rom Splines Combining B-splines and
Interpolation

• Dr. Scott Schaefer

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Disadvantages of B-splines

• B-splines dont interpolate vertices

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Disadvantages of Lagrange Interpolation

• Lagrange interpolation lacks local control

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Catmull-Rom Splines

• Given a set of points pk at parameter values tk,
  construct a curve C(t) such that C(tk)pk, C(t)
  is smooth and C(t) has local control
• Combining B-splines and Lagrange interpolation
  satisfies all of these properties!!

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Nevilles Algorithm
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B-Spline Curves
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Catmull-Rom Splines
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Catmull-Rom Splines
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Catmull-Rom Splines
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Catmull-Rom Splines
de Boor
Lagrange
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Catmull-Rom Splines
de Boor
Lagrange
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Catmull-Rom Spline Properties

• Piecewise polynomial of degree 2n-1
• Local control
• Interpolate pk at knots tk
• Cn-1 continuity at knots

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Catmull-Rom Spline Properties

• Piecewise polynomial of degree 2n-1
• Local control
• Interpolate pk at knots tk
• Cn-1 continuity at knots

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Catmull-Rom Spline Properties

• Piecewise polynomial of degree 2n-1
• Local control
• Interpolate pk at knots tk
• Cn-1 continuity at knots

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Catmull-Rom Spline Properties

• Piecewise polynomial of degree 2n-1
• Local control
• Interpolate pk at knots tk
• Cn-1 continuity at knots

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Catmull-Rom Spline Properties

• Piecewise polynomial of degree 2n-1
• Local control
• Interpolate pk at knots tk
• Cn-1 continuity at knots

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Catmull-Rom Spline Properties

• Piecewise polynomial of degree 2n-1
• Local control
• Interpolate pk at knots tk
• Cn-1 continuity at knots

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Smoothness of Catmull-Rom Splines

• Cn-1 continuity at knots

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Smoothness of Catmull-Rom Splines

• Cn-1 continuity at knots

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Smoothness of Catmull-Rom Splines

• Cn-1 continuity at knots

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Smoothness of Catmull-Rom Splines

• Cn-1 continuity at knots

de Boor
Lagrange
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Smoothness of Catmull-Rom Splines

• Cn-1 continuity at knots

de Boor
Lagrange
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Smoothness of Catmull-Rom Splines

• Cn-1 continuity at knots

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Smoothness of Catmull-Rom Splines

• Cn-1 continuity at knots

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Example
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Example