1 Curves and Surfaces. 2 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric Bi-Cubic Surfaces Quadric Surfaces Specialized.

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1 Curves and Surfaces

2 Representation of Curves & Surfaces 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

5 Pointers to a Vertex List

6 Pointers to an Edge List

7 Plane Equation Ax+By+Cz+D=0 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

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

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

10 Continuity between curve segments

11 Tangent Vector

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

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

14 Continuity between curve segments

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

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

18 Parametric Cubic Curves where is called the blending function

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 Hermite Curves Since

21 Hermite Curves so and

22 Bezier Curves Four control points 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, 4.9

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

24 Bézier Curves by definition then so

25 Bézier Curves and

26 Subdividing Bézier Curves How to draw the curve ? How to convert it to be line- segments ?

27 Bezier Curves Constructing a Bezier curve Fig. 4.10-13 Finding mid-points of lines

28 Bezier Curves

29 Convex Hull

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

31 B-Spline

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

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

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 Ex. (0,0,0,0,1,1,2,3,4,4,5,5,5,5) so

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

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

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

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

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

41 Parametric Bi-Cubic Surfaces so

42 Hermite Surfaces

43 Bézier Surfaces

44 B-Spline Surfaces

45 Normals to Surfaces

46 Quadric Surfaces implicit surface equation an alternative representation

47 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

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