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

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

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

13. Continuity Between Curve Segments

14. Continuity Between Curve Segments

15. Three Types of Parametric 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

16. 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

17. Parametric Cubic Curves
Where is called the blending functions

18. 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]

19. Hermite Curves
Since

20. Hermite Curves So
And

21. 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 matrix MB
Bézier geometry vector GB
Bézier blending functions BB

22. Bézier Curves
By definition
Then So

23. Bézier Curves
And
(Bernstein polynomials)

24. Convex Hull

25. 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

26. B-Spline

27. 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

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

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

30. 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

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

32. 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

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

34. NURBS: NonUniform Rational B-Splines
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

35. 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

36. Parametric Bi-Cubic SurfacesSo

37. Hermite Surfaces

38. Bézier Surfaces

39. Normals to Surfaces
normal vector

40. Quadric Surfaces Implicit surface equation
An alternative representation
with

41. Quadric Surfaces Advantages: Computing the surface normal
Testing whether a point is on the surface
Computing z given x and y
Calculating intersections of one surface with another

42. Fractal Models

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