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1 Curves and Surfaces 1

2 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized Modeling Techniques 2

3 The Teapot 3

4 Representing Polygon Meshes explicit representation by a list of vertex coordinates pointers to a vertex list pointers to an edge list 4

5 Pointers to a Vertex List 5

6 Pointers to an Edge List 6

7 Ax+By+Cz+D=0 Plane Equation And (A, B, C) means the normal vector so, given points P 1, P 2, and P 3 on the plane (A, B, C) =P 1 P 2 P 1 P 3 What happened if (A, B, C) =(0, 0, 0)? The distance from a vertex (x, y, z) to the plane is 7

8 8 Parametric Cubic Curves The cubic polynomials that define a curve segment are of the form

9 9 Parametric Cubic Curves The curve segment can be rewrite as where

10 10 Continuity between curve segments

11 Tangent Vector 11

12 Continuity between curve segments G 0 geometric continuity two curve segments join together G 1 geometric continuity the directions (but not necessarily the magnitudes) of the two segments tangent vectors are equal at a join point 12

13 Continuity between curve segments C 1 continuous the tangent vectors of the two cubic curve segments are equal (both directions and magnitudes) at the segments join point C n continuous the direction and magnitude of through the nth derivative are equal at the join point 13

14 14 Continuity between curve segments

15 15 Continuity between curve segments

16 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

17 Parametric Cubic Curves Rewrite the coefficient matrix as where M is a 4 4 basis matrix, G is called the geometry matrix so 17

18 18 Parametric Cubic Curves where function is called the blending

19 19 Hermite Curves Given the endpoints P 1 and P 4 and tangent vectors at R 1 and R 4 What are Hermite basis matrix M H Hermite geometry vector G H Hermite blending functions B H By definition

20 20 Hermite Curves Since

21 21 Hermite Curves so and

22 Four control points Bezier Curves Two endpoints, two direction points Length of lines from each endpoint to its direction point representing the speed with which the curve sets off towards the direction point Fig. 4.8,

23 Bézier Curves Given the endpoints and and two control points and which determine the endpoints tangent vectors, such that What is Bézier basis matrix M B Bézier geometry vector G B Bézier blending functions B B 23

24 by definition then Bézier Curves so 24

25 25 Bézier Curves and

26 Subdividing Bézier Curves How to draw the curve? How to convert it to be linesegments? 26

27 Bezier Curves Constructing a Bezier curve Fig Finding mid-points of lines 27

28 Bezier Curves 28

29 Convex Hull 29

30 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 30

31 B-Spline 31

32 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 32

33 Uniform NonRational B-Splines Curve segment Q i is defined by points thus B-Spline geometry matrix if then 33

34 34 Uniform NonRational B-Splines so B-Spline basis matrix B-Spline blending functions

35 NonUniform NonRational B-Splines the knot-value sequence is a nondecreasing sequence allow multiple knot and the number of identical parameter is the multiplicity so Ex. (0,0,0,0,1,1,2,3,4,4,5,5,5,5) 35

36 NonUniform NonRational B-Splines Where is the jth-order blending function for weighting control point p i 36

37 Knot Multiplicity & Continuity Since Q(t i ) is within the convex hull of P i-3, P i-2, and P i-1 If t i =t i+1, Q(t i ) is within the convex hull of P i-3, P i-2, and P i-1 and the convex hull of P i-2, P i-1, and P i, so it will lie on P i-2 P i-1 If t i =t i+1 =t i+2, Q(t i ) will lie on p i-1 If t i =t i+1 =t i+2 =t i+3, Q(t i ) will lie on both P i-1 and P i, and the curve becomes broken 37

38 Knot Multiplicity & Continuity multiplicity 1 : C 2 continuity multiplicity 2 : C 1 continuity multiplicity 3 : C 0 continuity multiplicity 4 : no continuity 38

39 NURBS: NonUniform Rational B-Splines rational x(t), y(t) and z(t) 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 39

40 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 G i (t) are cubics 40

41 41 Parametric Bi-Cubic Surfaces so

42 Hermite Surfaces 42

43 Bézier Surfaces 43

44 B-Spline Surfaces 44

45 Normals to Surfaces 45

46 46 Quadric Surfaces implicit surface equation an alternative representation

47 advantages Quadric Surfaces 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 47

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