1. 12/9/2016 02:28 UML Graphics II 91.547 B-Splines NURBS Session 3A

2. 22/9/2016 02:28 UML B-splines Suppose you wanted C 0, C 1 and C 2 continuity at curve boundaries. Use all four control points to determine boundary continuities and only require that the curve pass “close” to the points.

3. 32/9/2016 02:28 UML B-splines: Sharing of Control Points

4. 42/9/2016 02:28 UML B-splines: Using continuity requirements to compute geometry matrix/blending functions C 0 continuity here requires:

5. 52/9/2016 02:28 UML B-splines: Using continuity requirements to compute geometry matrix/blending functions

6. 62/9/2016 02:28 UML B-splines: Using continuity requirements to compute geometry matrix/blending functions Similarly, the C 1 and C 2 continuity conditions give:

7. 72/9/2016 02:28 UML B-spline blending functions 0 1

8. 82/9/2016 02:28 UML B-splines: Local versus global parameter

9. 92/9/2016 02:28 UML B-splines: Recursively defined basis functions For any “knot vector”: Order i

10. 102/9/2016 02:28 UML First order basis functions:

11. 112/9/2016 02:28 UML Second order basis functions:

12. 122/9/2016 02:28 UML Knot Vectors Only Requirement: Image: David Rogers

13. 132/9/2016 02:28 UML Definition of B Spline Curve Order of the spline Number of control points Number of knots in knot vector * * Notation according to D.F. Rogers

14. 142/9/2016 02:28 UML Knot Vectors: Open, Uniform Result: spline passes through end control vertices Image: David Rogers

15. 152/9/2016 02:28 UML Building Up Basis Functions Image: David Rogers

16. 162/9/2016 02:28 UML Methods of Control 0 Change number and/or position of control vertices 0 Change order k 0 Change type of knot vector -Open uniform -Open non uniform 0 Use multiple coincident control vertices 0 Use multiple internal knot values Image: David Rogers

17. 172/9/2016 02:28 UML Control: Change Order Image: David Rogers

18. 182/9/2016 02:28 UML Control: Non Uniform Knot Vectors Image: David Rogers

19. 192/9/2016 02:28 UML Control: Knot Vector Type Image: David Rogers

20. 202/9/2016 02:28 UML Control: Multiple Coincident Vertices Image: David Rogers

21. 212/9/2016 02:28 UML Control: Duplicate Knot Values Image: David Rogers

22. 222/9/2016 02:28 UML Rational B-Splines (NURBS) Equivalency between Homogeneous representations: Doing the perspective division gives: Interpreted as “weighting factor” for control vertices

23. 232/9/2016 02:28 UML NURBS Effect of weighting factor Image: David Rogers

24. 242/9/2016 02:28 UML Drawing NURBS in OpenGL GLUnurbsObj *curveName; curveName = gluNewNurbsRenderer(); gluBeginCurve (curveName); gluNurbsCurve (curveName, nknots, *knotVector, stride, *ctrlPts, degParam, GL_MAP1_VERTEX_3); gluEndCurve (curveName); See OpenGL Programming Guide Ch. 12 for details of using the glu NURBS interface

25. 252/9/2016 02:28 UML NURBS: Code Example 120 goto 120

26. 262/9/2016 02:28 UML Extending from Curves to Surfaces Cartesian product of B-Spline basis functions Order can be different for u and v directions