1 dr. scott schaefer catmull-rom splines: combining b-splines and interpolation
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![Page 1: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/1.jpg)
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Dr. Scott Schaefer
Catmull-Rom Splines: Combining B-splines and Interpolation
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Disadvantages of B-splines
B-splines don’t 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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Neville’s Algorithm
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
)(01 tP )(12 tP )(23 tP
)(012 tP )(123 tP
)(0123 tP
0tt tt 2 tt 3 1tt
tt 3 0tt
![Page 6: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/6.jpg)
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B-Spline Curves
m
i
nii tNptp
1
)()(
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Catmull-Rom Splines
m
inii
nin tPtNtC
1,...,
1 )()()(
![Page 8: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/8.jpg)
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Catmull-Rom Splines
m
inii
nin tPtNtC
1,...,
1 )()()(
1degree n ndegree12degree n
![Page 9: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/9.jpg)
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Catmull-Rom Splines
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
)(01 tP )(12 tP )(23 tP
)(012 tP )(123 tP
)(12 tC
0tt tt 2 tt 3 1tt
tt 2 1tt
tt 4 3tt
)(34 tP
)(234 tP
)(23 tC
tt 4 2tt
tt 3 2tt
4p
![Page 10: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/10.jpg)
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Catmull-Rom Splines
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
)(01 tP )(12 tP )(23 tP
)(012 tP )(123 tP
)(12 tC
0tt tt 2 tt 3 1tt
tt 4 3tt
)(34 tP
)(234 tP
)(23 tC
tt 4 2tt
4p
Lag
rang
ede
Boo
r
tt 2 1tt tt 3 2tt
![Page 11: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/11.jpg)
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Catmull-Rom Splines
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
)(01 tP )(12 tP )(23 tP
)(012 tP )(123 tP
)(0123 tP
0tt tt 2 tt 3 1tt
tt 3 0tt
tt 4 3tt
)(34 tP
)(234 tP
)(1234 tP
tt 4 2tt
tt 4 1tt
4p
tt 5 4tt
)(45 tP
)(345 tP
)(2345 tP
tt 5 3tt
tt 5 2tt
5p
tt 6 5tt
)(56 tP
)(456 tP
)(3456 tP
tt 6 4tt
tt 6 3tt
6p
)(123 tC )(234 tC
)(23 tC
)(345 tC
)(34 tC
Lag
rang
ede
Boo
r
tt 3 1tt tt 4 2tt
tt 3 2tt
tt 5 3tt
tt 4 3tt
![Page 12: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/12.jpg)
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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
m
inii
nin tPtNtC
1,...,
1 )()()(
![Page 14: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/14.jpg)
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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
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
)(01 tP )(12 tP )(23 tP
)(012 tP )(123 tP
)(12 tC
0tt tt 2 tt 3 1tt
tt 2 1tt
tt 4 3tt
)(34 tP
)(234 tP
)(23 tC
tt 4 2tt
tt 3 2tt
4p
![Page 15: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/15.jpg)
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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
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
)(01 tP )(12 tP )(23 tP
)(012 tP )(123 tP
)(12 tC
0tt tt 2 tt 3 1tt
tt 2 1tt
tt 4 3tt
)(34 tP
)(234 tP
)(23 tC
tt 4 2tt
tt 3 2tt
4p
![Page 16: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/16.jpg)
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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
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
? 2p 2p
2p 2p
2212 )( ptC
0tt tt 2 tt 3 1tt
tt 2 1tt
tt 4 3tt
?
2p
2223 )( ptC
tt 4 2tt
tt 3 2tt
4p
![Page 17: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/17.jpg)
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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
![Page 18: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/18.jpg)
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Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
1,...,
1 )()()(
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Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
1,...,
1 )()()(
C2nC
![Page 20: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/20.jpg)
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Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
1,...,
1 )()()(
C2nC
![Page 21: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/21.jpg)
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Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
)(01 tP )(12 tP )(23 tP
)(012 tP )(123 tP
)(0123 tP
0tt tt 2 tt 3 1tt
tt 3 0tt
tt 4 3tt
)(34 tP
)(234 tP
)(1234 tP
tt 4 2tt
tt 4 1tt
4p
tt 5 4tt
)(45 tP
)(345 tP
)(2345 tP
tt 5 3tt
tt 5 2tt
5p
tt 6 5tt
)(56 tP
)(456 tP
)(3456 tP
tt 6 4tt
tt 6 3tt
6p
)(123 tC )(234 tC
)(23 tC
)(345 tC
)(34 tC
Lag
rang
ede
Boo
r
tt 3 1tt tt 4 2tt
tt 3 2tt
tt 5 3tt
tt 4 3tt
![Page 22: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/22.jpg)
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Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
0p 1p 2p 3p
tt 1 0tt tt 2 tt 31tt 2tt
)(01 tP )(12 tP )(23 tP
)(012 tP )(123 tP
)(0123 tC
0tt tt 2 tt 3 1tt
tt 3 0tt
tt 4 3tt
)(34 tP
)(234 tP
)(1234 tC
tt 4 2tt
tt 4 1tt
4p
tt 5 4tt
)(45 tP
)(345 tP
)(2345 tC
tt 5 3tt
tt 5 2tt
5p
tt 6 5tt
)(56 tP
)(456 tP
)(3456 tC
tt 6 4tt
tt 6 3tt
6p
)(123 tCtt 3 1tt
)(234 tC
)(23 tC
tt 4 2tt
tt 3 2tt
)(345 tC
)(34 tC
tt 5 3tt
tt 4 3tt
Lag
rang
ede
Boo
r
![Page 23: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/23.jpg)
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Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
1,...,
1 )()()(
C2nC
![Page 24: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/24.jpg)
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Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
11,..., )()()(
C1nC
![Page 25: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/25.jpg)
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Example
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Example
![Page 27: 1 Dr. Scott Schaefer Catmull-Rom Splines: Combining B-splines and Interpolation](https://reader035.vdocuments.us/reader035/viewer/2022062216/56649d5c5503460f94a3b244/html5/thumbnails/27.jpg)
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Example
Uniform Parameterization
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Example
Centripetal Parameterization
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Example
Chordal Parameterization