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Dr. Scott Schaefer
Catmull-Rom Splines: Combining B-splines and Interpolation
2/29
Disadvantages of B-splines
B-splines don’t interpolate vertices
3/29
Disadvantages of Lagrange Interpolation Lagrange interpolation lacks local control
4/29
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!!
5/29
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
6/29
B-Spline Curves
m
i
nii tNptp
1
)()(
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Catmull-Rom Splines
m
inii
nin tPtNtC
1,...,
1 )()()(
8/29
Catmull-Rom Splines
m
inii
nin tPtNtC
1,...,
1 )()()(
1degree n ndegree12degree n
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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
10/29
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
11/29
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
12/29
Catmull-Rom Spline Properties
Piecewise polynomial of degree 2n-1 Local control Interpolate pk at knots tk Cn-1 continuity at knots
13/29
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 )()()(
14/29
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
15/29
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
16/29
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
17/29
Catmull-Rom Spline Properties
Piecewise polynomial of degree 2n-1 Local control Interpolate pk at knots tk Cn-1 continuity at knots
18/29
Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
1,...,
1 )()()(
19/29
Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
1,...,
1 )()()(
C2nC
20/29
Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
1,...,
1 )()()(
C2nC
21/29
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
22/29
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
23/29
Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
1,...,
1 )()()(
C2nC
24/29
Smoothness of Catmull-Rom Splines
Cn-1 continuity at knots
m
inii
nin tPtNtC
11,..., )()()(
C1nC
25/29
Example
26/29
Example
27/29
Example
Uniform Parameterization
28/29
Example
Centripetal Parameterization
29/29
Example
Chordal Parameterization
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