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

2. 2/29 Disadvantages of B-splines B-splines don’t interpolate vertices

3. 3/29 Disadvantages of Lagrange Interpolation Lagrange interpolation lacks local control

4. 4/29 Catmull-Rom Splines Given a set of points p k at parameter values t k, construct a curve C(t) such that C(t k )=p k, C(t) is smooth and C(t) has local control Combining B-splines and Lagrange interpolation satisfies all of these properties!!

5. 5/29 Neville’s Algorithm

6. 6/29 B-Spline Curves

7. 7/29 Catmull-Rom Splines

8. 8/29 Catmull-Rom Splines

9. 9/29 Catmull-Rom Splines

10. 10/29 Catmull-Rom Splines Lagrange de Boor

11. 11/29 Catmull-Rom Splines Lagrange de Boor

12. 12/29 Catmull-Rom Spline Properties Piecewise polynomial of degree 2n-1 Local control Interpolate p k at knots t k C n-1 continuity at knots

13. 13/29 Catmull-Rom Spline Properties Piecewise polynomial of degree 2n-1 Local control Interpolate p k at knots t k C n-1 continuity at knots

14. 14/29 Catmull-Rom Spline Properties Piecewise polynomial of degree 2n-1 Local control Interpolate p k at knots t k C n-1 continuity at knots

15. 15/29 Catmull-Rom Spline Properties Piecewise polynomial of degree 2n-1 Local control Interpolate p k at knots t k C n-1 continuity at knots

16. 16/29 Catmull-Rom Spline Properties Piecewise polynomial of degree 2n-1 Local control Interpolate p k at knots t k C n-1 continuity at knots

17. 17/29 Catmull-Rom Spline Properties Piecewise polynomial of degree 2n-1 Local control Interpolate p k at knots t k C n-1 continuity at knots

18. 18/29 Smoothness of Catmull-Rom Splines C n-1 continuity at knots

19. 19/29 Smoothness of Catmull-Rom Splines C n-1 continuity at knots

20. 20/29 Smoothness of Catmull-Rom Splines C n-1 continuity at knots

21. 21/29 Smoothness of Catmull-Rom Splines C n-1 continuity at knots Lagrange de Boor

22. 22/29 Smoothness of Catmull-Rom Splines C n-1 continuity at knots Lagrange de Boor

23. 23/29 Smoothness of Catmull-Rom Splines C n-1 continuity at knots

24. 24/29 Smoothness of Catmull-Rom Splines C n-1 continuity at knots

25. 25/29 Example

26. 26/29 Example

27. 27/29 Example Uniform Parameterization

28. 28/29 Example Centripetal Parameterization

29. 29/29 Example Chordal Parameterization