Page 1 CS148: Introduction to Computer Graphics and Imaging Splines and Curves CS148 Lecture 9 Pat Hanrahan, Winter 2009 Topics Splines Cubic Hermite interpolation Matrix representation of cubic polynomials Catmull-Rom interpolation Curves Bezier curve Chaiken’s subdivision algorithm Properties of Bezier curves
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Page 1
CS148: Introduction to Computer Graphics and Imaging
Splines and Curves
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Topics
Splines
Cubic Hermite interpolation
Matrix representation of cubic polynomials
Catmull-Rom interpolation
Curves
Bezier curve
Chaiken’s subdivision algorithm
Properties of Bezier curves
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Cubic Hermite Interpolation
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Cubic Hermite Interpolation
Given: values and derivatives at 2 points
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Assume cubic polynomial
Solve for coefficients:
Cubic Hermite Interpolation
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Matrix Representation
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Matrix Representation
Inverse
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Matrix Representation
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Matrix Representation of Polynomials
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Matrix Representation of Polynomials
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Matrix Representation of Polynomials
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Matrix Representation
Transpose
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Matrix Representation of Polynomials
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Hermite Basis Matrix
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Hermite Basis Matrix
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Hermite Basis Functions
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Catmull-Rom Interpolation
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Catmull-Rom Interpolation
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Catmull-Rom Interpolation
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Catmull-Rom Interpolation
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Catmull-Rom Interpolation
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Catmull-Rom Interpolation
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Catmull-Rom Interpolation
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Catmull-Rom To Hermite Interpolation
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Bezier Curves
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Paths
Capabilities
1. Smooth curves
2. Line and curve segments 3. Kinks
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier To Hermite Interpolation
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Demo of Bezier Curves
Subdivision
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Chaiken’s Algorithm
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Chaiken’s Algorithm
Page 17
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Chaiken’s Algorithm
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Chaiken’s Algorithm
Page 18
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Chaiken’s Algorithm
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve
Page 19
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Properties
Property 1: Interpolate end points
Property 2: Tangents
Property 3: Convex hull property
Extrapolation
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve - Extrapolation
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve - Extrapolation
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve - Extrapolation
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve - Extrapolation
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve
Left Bezier Curve
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bezier Curve
Right Bezier Curve Left Bezier Curve
Can subdivide a Bezier curve into two pieces
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Applications of Subdivision
Drawing Bezier curve
??
Intersect two Bezier curves
??
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Continuity between 2 Bezier Curves
3rd point of the 1st curve is the same as the 1st point of
the 2nd curve
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Continuity between 2 Bezier Curves
Tangent of the 1st curve is equal to the tangent of the
2nd curve
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Pyramid Algorithm
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Pyramid Algorithm
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Pyramid Algorithm
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Pyramid Algorithm
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Pyramid Algorithm
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Bernstein Polynomials
CS148 Lecture 9 Pat Hanrahan, Winter 2009
Pyramid Algorithm
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CS148 Lecture 9 Pat Hanrahan, Winter 2009
Things to Remember
Splines
Cubic hermite interpolation
Matrix representation of cubic polynomials
Catmull-Rom splines
How to think of CR in terms of Hermite spline
Curves
Bezier curve
How to think of BC in terms of Hermite spline
Chaiken’s algorithm
Subdivision algorithm including applications
Properties of Bezier curves
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