Representation of Curves & Surfaces

1. Representation of Curves & Surfaces Prof. Lizhuang Ma Shanghai Jiao Tong University

2. Contents • Specialized Modeling Techniques • Polygon Meshes • Parametric Cubic Curves • Parametric Bi-Cubic Surfaces • Quadric Surfaces • Specialized Modeling Techniques

3. The Teapot

4. Representing Polygon Meshes • Explicit representation • By a list of vertex coordinates • Pointers to a vertex list • Pointers to an edge list 1K factes 10K factes

5. Pointers to A Vertex List

6. Pointers to An Edge List

7. Parametric Cubic Curves • The cubic polynomials that define a curve segment are of the form:

8. Parametric Cubic Curves • The curve segment can be rewritten as • Where

9. Parametric Cubic Curves

10. Tangent Vector

11. ContinuityBetween Curve Segments • G0 geometric continuity • Two curve segments join together • G1 geometric continuity • The directions (but not necessarily the magnitudes) of the two segments’ tangent vectors are equal at a join point

12. ContinuityBetween Curve Segments • C1 continuous • The tangent vectors of the two cubic curve segments are equal (both directions and magnitudes) at the segments’ joint point􀂆 • Cncontinuous • The direction and magnitude of through the nth derivative are equal at the joint point

13. Examples • C1 continuous • "looks smooth, no facets" • C2 continuous • Actually important for shading

14. ContinuityBetween Curve Segments

15. ContinuityBetween Curve Segments

16. Three Types ofParametric Cubic Curves • Hermite Curves • Defined by two endpoints and two endpoint tangent vectors • Bézier Curves • Defined by two endpoints and two control points which control the endpoint’ tangent vectors • Splines • Defined by four control points

17. Parametric Cubic Curves • Representation: • Rewrite the coefficient matrix as • where M is a 4x4 basis matrix, G is called the geometry matrix • So 4 endpoints or tangent vectors

18. Parametric Cubic Curves • Where is called the blending functions

19. Hermite Curves • Given the endpoints P1 and P4 and tangent vectors at them R1 and R4 • What is • Hermite basis matrix MH • Hermite geometry vector GH • Hermite blending functions BH • By definition GH=[P1 P4 R1 R4]

20. Hermite Curves • Since

21. Hermite Curves • So • And

22. Bézier Curves • Given the endpoints P1 and P4 and two control points P2 and P3 which determine the endpoints’ tangent vectors, such that • What is􀂄 • Bézier basis matrixMB • Bézier geometry vector GB • Bézier blending functions BB

23. Bézier Curves • By definition • Then • So

24. Bézier Curves • And (Bernstein polynomials)

25. Convex Hull

26. Spline • The polynomial coefficients for natural cubic splines are dependent on all n control points • Has one more degree of continuity than is inherent in the Hermite and Bézier forms • Moving any one control point affects the entire curve • The computation time needed to invert the matrix can interfere with rapid interactive reshaping of a curve

27. B-Spline

28. Uniform NonRational B-Splines • Cubic B-Spline • Has m+1 control points • Has m-2 cubic polynomial curve segments • Uniform • The knots are spaced at equal intervals of the parameter t • Non-rational • Not rational cubic polynomial curves

29. Uniform NonRational B-Splines • Curve segment is defined by points • B-Spline geometry matrix • If • Then

30. Uniform NonRational B-Splines • So B-Spline basis matrix • B-Spline blending functions

31. Uniform NonRational B-Splines • The knot-value sequence is a nondecreasing sequence • Allow multiple knot and the number of identical parameter is the multiplicity • Ex. (0,0,0,0,1,1,2,3,4,4,5,5,5,5) • So

32. NON-Uniform NonRational B-Splines • Where is j th-order blending function for weighting control point Pi

33. Knot Multiplicity & Continuity • Since is within the convex hull of and • If is within the convex hull of , and and the convex hull of and ,so it will lie on • If will lie on • If will lie on both and , and the curve becomes broken

34. Knot Multiplicity & Continuity • Multiplicity 1 : C2 continuity • Multiplicity 2 : C1 continuity • Multiplicity 3 : C0 continuity • Multiplicity 4 : no continuity

35. NURBS:NonUniform Rational B-Splines • Rational • and are defined as the ratio of two cubic polynomials • Rational cubic polynomial curve segments are ratios of polynomials • Can be Bézier, Hermite, or B-Splines

36. Parametric Bi-Cubic Surfaces • Parametric cubic curves are • So, parametric bi-cubic surfaces are • If we allow the points in G to vary in 3D along some path, then • Since are cubics: • , where

37. Parametric Bi-Cubic Surfaces • So

38. Hermite Surfaces

39. Bézier Surfaces

40. Normals to Surfaces normal vector

41. Quadric Surfaces • Implicit surface equation • An alternative representation • with

42. Quadric Surfaces • Advantages: • Computing the surface normal • Testing whether a point is on the surface • Computing zgiven xand y • Calculating intersections of one surface with another

43. Fractal Models

44. Grammar-Based Models • L-grammars • A -> AA • B -> A[B]AA[B]