Curves and Surfaces 1


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1 Curves and Surfaces 1
2 Representation of Curves & Surfaces Polygon Meshes Parametric Cubic Curves Parametric BiCubic 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 midpoints 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 BSpline 31
32 Uniform NonRational BSplines cubic BSpline has m+1 control points has m2 cubic polynomial curve segments uniform the knots are spaced at equal intervals of the parameter t nonrational not rational cubic polynomial curves 32
33 Uniform NonRational BSplines Curve segment Q i is defined by points thus BSpline geometry matrix if then 33
34 34 Uniform NonRational BSplines so BSpline basis matrix BSpline blending functions
35 NonUniform NonRational BSplines the knotvalue 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 BSplines Where is the jthorder blending function for weighting control point p i 36
37 Knot Multiplicity & Continuity Since Q(t i ) is within the convex hull of P i3, P i2, and P i1 If t i =t i+1, Q(t i ) is within the convex hull of P i3, P i2, and P i1 and the convex hull of P i2, P i1, and P i, so it will lie on P i2 P i1 If t i =t i+1 =t i+2, Q(t i ) will lie on p i1 If t i =t i+1 =t i+2 =t i+3, Q(t i ) will lie on both P i1 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 BSplines 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 BSplines 39
40 Parametric BiCubic Surfaces Parametric cubic curves are so parametric bicubic 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 BiCubic Surfaces so
42 Hermite Surfaces 42
43 Bézier Surfaces 43
44 BSpline 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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