CatmullRom Splines Combining Bsplines and Interpolation Dr Scott

Catmull-Rom Splines: Combining B-splines and Interpolation Dr. Scott Schaefer 1

Disadvantages of B-splines n B-splines don’t interpolate vertices 2

Disadvantages of Lagrange Interpolation n Lagrange interpolation lacks local control 3

Catmull-Rom Splines n 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 n Combining B-splines and Lagrange interpolation satisfies all of these properties!! 4

Neville’s Algorithm 5

B-Spline Curves 6

Catmull-Rom Splines 7

Catmull-Rom Splines 8

Catmull-Rom Splines 9

Lagrange de Boor Catmull-Rom Splines 10

Lagrange de Boor Catmull-Rom Splines 11

Catmull-Rom Spline Properties Piecewise polynomial of degree 2 n-1 n Local control n Interpolate pk at knots tk n Cn-1 continuity at knots n 12

Catmull-Rom Spline Properties Piecewise polynomial of degree 2 n-1 n Local control n Interpolate pk at knots tk n Cn-1 continuity at knots n 13

Catmull-Rom Spline Properties Piecewise polynomial of degree 2 n-1 n Local control n Interpolate pk at knots tk n Cn-1 continuity at knots n 14

Catmull-Rom Spline Properties Piecewise polynomial of degree 2 n-1 n Local control n Interpolate pk at knots tk n Cn-1 continuity at knots n 15

Catmull-Rom Spline Properties Piecewise polynomial of degree 2 n-1 n Local control n Interpolate pk at knots tk n Cn-1 continuity at knots n 16

Catmull-Rom Spline Properties Piecewise polynomial of degree 2 n-1 n Local control n Interpolate pk at knots tk n Cn-1 continuity at knots n 17

Smoothness of Catmull-Rom Splines n Cn-1 continuity at knots 18

Smoothness of Catmull-Rom Splines n Cn-1 continuity at knots 19

Smoothness of Catmull-Rom Splines n Cn-1 continuity at knots 20

Smoothness of Catmull-Rom Splines Cn-1 continuity at knots Lagrange de Boor n 21

Smoothness of Catmull-Rom Splines Cn-1 continuity at knots Lagrange de Boor n 22

Smoothness of Catmull-Rom Splines n Cn-1 continuity at knots 23

Smoothness of Catmull-Rom Splines n Cn-1 continuity at knots 24

Example 25

Example 26

Example Uniform Parameterization 27

Example Centripetal Parameterization 28

Example Chordal Parameterization 29