1. Parametric Surfaces January 16, 2003 Stephen Gordon

2. Outline Introduction Fundamentals –Parametric Curves Bézier B-Spline Parametric Surfaces –Usage –Applications Current Trends

3. What are the Parametric Advantages? Provides exact analytical representation Allows 3D shape editing More economical

4. Why Backseat to Polygon Mesh? Extensive mathematics Overkill for many applications Currently experiencing an evolution.

5. Where is it Used? CAD interactive design –Representing real objects Entertainment –Movies –Video games

6. Fundamentals – Bézier Curves Pierre Bézier created UNISURF in 1960’s for automotive design at Renault.

7. Fundamentals – Bézier Curves P 0, P 1, & P 2 are control points. Q(t) is interpolated between 0 and 1.

8. Fundamentals – B-Spline Curves Generalization of Bézier Curve Sequence of control points that guarantee continuity.

9. Bézier Vs. B-Spline Bézier –Less computation B-Spline –Exhibits non-localness, result smoother –Multiple curve segments not necessary

10. Bézier Patches Combine two Bézier curves to create a surface 16 control points

11. Bézier Patches Great for single patch surfaces Problems with multi-patch surfaces –“Cracking” can occur If adjacent patches are tessellated to different levels –To prevent, common edges must have matching tangent vectors

12. The Utah Teapot Bézier: 32 patches x 16 control points/patch = 288 vertices Polygon Mesh= 2048 vertices

13. B-Spline Patches Combination of 2 B-Spline curves 16 control points necessary

14. Bézier Vs. B-Spline 2 Bézier –Less computation B-Spline –Exhibits non-localness, result smoother –Multiple curve segments not necessary

15. What are Some Bézier Applications? Rough collision detection –Contained within convex hull of control points

16. What are Some Bézier Applications? Terrain rendering –Very good compression –Maintain constant frame rate Quake III uses Bézier patches to render the demonic tongue

17. More Terrain Rendering Shots below from SSX –Demonstrate versatility of Bézier patches

18. How are Models Created? Cross-sectional / linear axis design –Provides symmetry –Example: Vase Profile

19. How Else? Control polyhedron design –Modify control point and 8 neighbors Continuity is maintained –Fine control Control scale by subdividing –Coarse control Global deformation by changing curve shape

20. How Else? Surface fitting –Fit curves to 3D surface data points –Create curve network through interpolation Action figure Dense polygon mesh With curve network B-Spline Model

21. What About Bézier Triangles? Similar to Bézier patches –Not as straightforward –Used to form N-Patches Control Points of Cubic Bézier Triangle

22. So What are N-Patches? A triangular Bézier surface Adds detail to existing polygon mesh models –Better surface lighting –More realistic silhouette edges –Improves shape cheaply

23. Why are They Useful? Hardware support –Graphics cards can: Enable/disable NP’s Determine level of tessellation A more advanced technique curved NP Triangles are applied to these id Software models:

24. Recap Parametric surface advantages: Provides exact analytical representation Allows 3D shape editing More economical