1 Dr. Scott Schaefer Tensor-Product Surfaces

2/64 Smooth Surfaces Lagrange Surfaces  Interpolating sets of curves Bezier Surfaces B-spline Surfaces

3/64 Lagrange Surfaces

4/64 Lagrange Surfaces

5/64 Lagrange Surfaces

6/64 Lagrange Surfaces

7/64 Lagrange Surfaces

8/64 Lagrange Surfaces

9/64 Lagrange Surfaces

10/64 Lagrange Surfaces

11/64 Lagrange Surfaces

12/64 Lagrange Surfaces

13/64 Lagrange Surfaces

14/64 Lagrange Surfaces

15/64 Lagrange Surfaces

16/64 Lagrange Surfaces

17/64 Lagrange Surfaces – Properties Surface interpolates all control points The boundaries of the surface are Lagrange curves defined by the control points on the boundary

18/64 Interpolating Sets of Curves Given a set of parametric curves p 0 (t), p 1 (t), …, p n (t), build a surface that interpolates them

19/64 Interpolating Sets of Curves Given a set of parametric curves p 0 (t), p 1 (t), …, p n (t), build a surface that interpolates them Evaluate each curve at parameter value t, then use these points as the control points for a Lagrange curve of degree n Evaluate this new curve at parameter value s

20/64 Bezier Surfaces

21/64 Bezier Surfaces

22/64 Bezier Surfaces

23/64 Bezier Surfaces

24/64 Bezier Surfaces

25/64 Bezier Surfaces

26/64 Bezier Surfaces

27/64 Bezier Surfaces

28/64 Bezier Surfaces

29/64 Bezier Surfaces

30/64 Bezier Surfaces

31/64 Bezier Surfaces – Properties Surface lies in convex hull of control points Surface interpolates the four corner control points Boundary curves are Bezier curves defined only by control points on boundary

32/64 General Tensor Product Surfaces

33/64 General Tensor Product Surfaces

34/64 Properties Curve properties/algorithms apply to surfaces too  Convex hull

35/64 Properties Curve properties/algorithms apply to surfaces too  Convex hull  Degree elevation

36/64 Properties Curve properties/algorithms apply to surfaces too  Convex hull  Degree elevation  Evaluation algorithms

37/64 Properties Curve properties/algorithms apply to surfaces too  Convex hull  Degree elevation  Evaluation algorithms  ….  Analog of variation diminishing does not apply!!!

38/64 Matrix Form of Quadrilateral Bezier Patches

39/64 Matrix Form of Quadrilateral Bezier Patches

40/64 deCasteljau Algorithm for Bezier Surfaces

41/64 deCasteljau Algorithm for Bezier Surfaces

42/64 deCasteljau Algorithm for Bezier Surfaces

43/64 deCasteljau Algorithm for Bezier Surfaces

44/64 deCasteljau Algorithm for Bezier Surfaces

45/64 deCasteljau Algorithm for Bezier Surfaces

46/64 deCasteljau Algorithm for Bezier Surfaces

47/64 deCasteljau Algorithm for Bezier Surfaces

48/64 deCasteljau Algorithm for Bezier Surfaces

49/64 Derivatives of Bezier Surfaces Exact evaluate in the s-direction and use those control points to compute derivative in t-direction Exact evaluate in the t-direction and use those control points to compute derivative in s-direction Use a pyramid algorithm to compute derivatives

50/64 Derivatives using deCasteljau’s algorithm

51/64 Derivatives using deCasteljau’s algorithm

52/64 Derivatives using deCasteljau’s algorithm

53/64 Blossoming for Tensor-Product Patches Symmetry: b(s 1,s 2,…,s m |t 1,t 2,…,t n ) = b(s q(1),s q(2),…,s q(m) |t r(1),t r(2),…,t r(n) ) for any permutation q of (1,…,m) and r of (1,...,n) Multi-affine: b(s 1,…,(1-d)s k +d v k,,…s m |t 1,…,(1-e)t j +e w j,,…t n ) = (1-d)(1-e) b(s 1,…,s k,,…s m |t 1,…,t j,,…t n ) + (1-d)e b(s 1,…,s k,,…s m |t 1,…,w j,,…t n ) + de b(s 1,…,v k,,…s m |t 1,…,w j,,…t n ) + d(1-e) b(s 1,…,v k,,…s m |t 1,…,t j,,…t n ) Diagonal: b(s,s,…,s|t,t,…,t) = p(s,t)

54/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

55/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

56/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

57/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

58/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

59/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points.

60/64 Curves on Bezier Surfaces Assume we have points p(s 1,t 1 ), p(s 2,t 2 ) on a Bezier surface p(s,t). Construct a curve on the Bezier surface between these two points. k th control point for Bezier curve of degree n+m!!!

61/64 Triangular Patches How do we build triangular patches instead of quads?

62/64 Triangular Patches How do we build triangular patches instead of quads?

63/64 Triangular Patches How do we build triangular patches instead of quads?

64/64 Triangular Patches How do we build triangular patches instead of quads? Continuity difficult to maintain between patches Parameterization very distorted Not symmetric