1. Surfaces Chiew-Lan Tai

2. Surfaces 2 Reading Required Hills Section 11.11 Hearn & Baker, sections 8.11, 8.13 Recommended Sections 2.1.4, 3.4-3.5, 3D Computer Graphics, Watt

3. Surfaces 3 Mathematical surface representations Explicit z = f(x,y) (a.k.a. “height field”) What if the surface isn’t a function? Implicit g(x,y,z) = 0 x 2 + y 2 + z 2 = 1 Parametric S(u,v) = (x(u,v), y(u,v), z(u,v)) x(u,v) = r cos 2  v sin  u y(u,v) = r sin 2  v sin  u z(u,v) = r cos  u 0≤u≤1, 0≤v≤1

4. Surfaces 4 Tensor product Bézier surfaces Given a grid of control point V ij, forming a control net, construct a surface S(u,v) by: Each row in u-direction is control points of a curve V i (u). V 0 (u), …, V n (u) at a specific u are the control points of a curve parameterized by v (i.e. S(u,.v))

5. Surfaces 5 Tensor product Bezier surfaces, cont. Let’s walk through the steps: Which points are interpolated by the surface? V 0 (u) and V 3 (u)

6. Surfaces 6 Tensor product Bezier surfaces, cont. Writing it out explicitly: Linear Bezier surface: Quadratic Bezier surface?

7. Surfaces 7 Matrix form For the case of cubic Bezier surface:

8. Surfaces 8 Tensor product B-spline surfaces Like spline curves, we can piece together a sequence of Bezier surfaces to make a spline surface. If we enforce C 2 continuity and local control, we get B-spline curves:

9. Surfaces 9 Tensor product B-spline surfaces Use each row of B control points in u to generate Bezier control points in u. Treat each row of Bezier control points in v direction as B-spline control points to generate Bezier control points in v direction. u

10. Surfaces 10 Tangent plane and surface normal S S

11. Surfaces 11 Partial Derivatives

12. Surfaces 12 Surface normal S SS S

13. Surfaces 13 Other construction methods

14. Surfaces 14 Rotational surfaces (surface of revolution) Rotate a 2D profile curve about an axis

15. Surfaces 15 Profile curve, C(u) Constructing surfaces of revolution Given a curve C(u) in the yz-plane: Let R y (  ) be a rotation about the y-axis Find: A surface S(u,v) obtained by applying R y (  ) on C(u) S(u,v) = R y (v) [0, C y (u), C z (u), 1] t y xz

16. Surfaces 16 Surface of revolution S(u,v) = ( C z (u) sin(v), C y (u), C z (u) cos(v)) y z x

17. Surfaces 17 General sweep surface Given a planar profile curve C(u) and a general space curve T(v) as the trajectory, sweep C(u) along the trajectory How to orient C(u) as it moves along T(v)?

18. Surfaces 18 Orientating C(u) Define a local coordinate frame at any point along the trajectory The Frenet frame (t,n,b) is the natural choice As we move along T(v), the Frenet frame (t,b,n) varies smoothly (inflection points where curvature goes to zero needs special treatment)

19. Surfaces 19 Sweep Surfaces Orient the profile curve C(u) using the Frenet frame of the trajectory T(v): Put C(u) in the normal plane. Place O c at T(v). Align x c of C(u) with b. Align y c of C(u) with n. Sweep surface S(u,v) = T(v) + n(v)C x (u) + b(v) C y (u)

20. Surfaces 20 Variations Several variations are possible: Scale C(u) as it moves, possibly using length of T(v) as a scale factor. Morph C(u) into some other curve C’(u) as it moves along T(v)

21. Surfaces 21 Summary What to take home: How to construct tensor product Bezier surfaces How to construct tensor product B-spline surfaces How to construct surfaces of revolution How to construct sweep surfaces from a profile and a trajectory curve with a Frenet frame

22. Surfaces 22 Subdivision Surfaces What’s wrong with B-spline/NURBS surfaces?

23. Surfaces 23 Subdivision curves Idea: repeatedly refine the control polygon curve is the limit of an infinite process.

24. Surfaces 24 Subdivision curves

25. Surfaces 25 Chaikin’s algorithm In 1974, Chaikin introduced the following “corner- cutting” scheme: Start with a piecewise linear curve Repeat Insert new vertices at the midpoints (the splitting step) Average two neighboring vertices (the average step) Averaging mask (0.5, 0.5) With this averaging mask, in the limit, the resulting curve is a quadratic B- spline curve.

26. Surfaces 26 Local subdivision mask Subdivision mask: ¼ (1, 2, 1) Splitting and averaging: Applying the step recursively, each vertex converges to a specific point With this mask, in the limit, the resulting curve is a cubic B-spline curve.

27. Surfaces 27 General subdivision process After each split-average step, we are closer to the limit surface. Can we push a vertex to its limit position without infinite subdivision? Yes! We can determine the final position of a vertex by applying the evaluation mask. The evaluation mask for cubic B-spline is 1/6 (1,4, 1)

28. Surfaces 28 DLG interpolating scheme (1987) Algorithm: splitting step introduces midpoints, and averaging step only changes these midpoints Dyn-Levin-Gregory scheme:

29. Surfaces 29 Building Complex Models This simple idea can be extended to build subdivision surfaces.

30. Surfaces 30 Subdivision surfaces Iteratively refine a control polyhedron (or control mesh) to produce the limit surface using splitting and averaging step: There are two types of splitting schemes: vertex schemes face schemes

31. Surfaces 31 Vertex schemes A vertex surrounded by n faces is split into n subvertices, one for each face: Doo-Sabin subdivision:

32. Surfaces 32 Face schemes Each quadrilateral face is split into four subfaces: Catmull-Clark subdivision:

33. Surfaces 33 Face scheme, cont. Each triangular face is split into four subfaces: Loop subdivision:

34. Surfaces 34 Averaging step Averaging masks:

35. Surfaces 35 Adding creases Sometimes, a particular feature such as a crease should be preserved. we just modify the subdivision mask. This gives rise to G 0 continuous surfaces.

36. Surfaces 36 Creases Here’s an example using Catmull-Clark surfaces: