CHAPTER 10 Curves and Surfaces

1. CHAPTER 10 Curves and Surfaces by Richard S. Wright Jr. • Vivian

3. Objectives • Introduce OpenGL evaluators • Learn to render polynomial curves and surfaces • Discuss quadrics in OpenGL - GLUT Quadrics - GLU Quadrics

4. What Does OpenGL Support? • Evaluators: a general mechanism for working with the Bernstein polynomials - Can use any degree polynomials - Can use in 1-4 dimensions - Automatic generation of normals and texture coordinates - NURBS supported in GLU • Quadrics • GLU and GLUT contain polynomial approximations of quadrics

5. Quadrics

6. Quadrics • Void gluQuadricDrawStyle(GLUquadricObj *obj, GLenum drawStyle); • void gluQuadricNormals(GLUquadricObj *pbj, GLenum normals);

7. Draw a quadrics • Draw a sphere: • void gluSphere(GLUQuadricObj *obj, GLdouble radius, GLint slices, GLint stacks);

8. Draw a quadrics void gltDrawUnitAxes(void)

9. Curves and Surfaces Overview • What is a parametric curve/surface? • Why use parametric curves & surfaces? • Bézier curves & surfaces • NURBS • Trimmed surfaces • OpenGL library

10. What is a parametric curve? 2D parametric curve takes the form y = mx + b x y f(t) g(t) Where f(t) and g(t) are functions of t = Example: Line thru points a and b (1-t) ax + t bx (1-t) ay+ t by x y = Mapping of the real line to 2D: here t in [0,1]  line segment a,b

11. What is a parametric curve? 3D curves defined similarly x y z f(t) g(t) h(t) = Example: helix x y z cos(t) sin(t) t =

12. Parametric representation of curves and surfaces.

13. Control Points The order of the curve is represented by the number of control points used to describe its shape. The degree is one less than the order of the curve.

14. Bézier Curves Examples linear: b(t) = (1-t) b0 + t b1 n=1 quadratic: b(t) = (1-t)2b0 + 2(1-t)t b1 + t2b2 n=2 cubic: b(t) = (1-t)3b0 + 3(1-t)2 t b1 + 3(1-t)t2b2 + t3b3 n=3 Bernstein basis Bin (t) = {n!/(n-i)! i!} (1-t)n-i ti

15. Continuity

16. Bézier Curves in OpenGL Basic steps: • Define curve by specifying degree, control points and parameter space [u0,u1] • Enable evaluator • Call evaluator with parameter u in [u0, u1] glMap1*() Autocreate uniformly spaced u:glMapGrid1*() glEvalMesh1() Specify each u: glEvalCoord1*() or Color and texture available too!

17. What is a parametric surface? 3D parametric surface takes the form Where f,g,h are bivariate functions of u and v x y z f(u,v) g(u,v) h(u,v) = Example: x(u,v) = u v u2 + v2 mapping u,v-space to 3-space; this happens to be a function too

18. Bézier Surfaces in OpenGL Basic steps: • Define curve by specifying degree, control points and parameter space [u0,u1] • Enable evaluator • Call evaluator with parameter u in [u0, u1,v0 , v1] glMap2*()

19. Bézier Surface Multiple patches connected smoothly Conditions on control netsimilar to curves … difficult to do manually

20. NURBS Non-uniform Rational B-splines B-splines are piecewise polynomials One or more Bezier curves /surfaces One control polygon Rational: let’s us represent circles exactly GLU NURBS utility

21. From Bézier to B-Splines

22. Creating a NURBS Surface • Evaluator →NURBS • Knots

23. Trimming

24. Trimmed Surfaces Parametric surface with parts of the domain “invisible” Surf Lab domain Jorg Peters’ UFL group GLU Trimmed NURBS utility