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1 Curves and Surfaces

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

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3 The Teapot

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

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5 Pointers to a Vertex List

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6 Pointers to an Edge List

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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

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

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

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10 Continuity between curve segments

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11 Tangent Vector

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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

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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

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14 Continuity between curve segments

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15 Continuity between curve segments

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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

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

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18 Parametric Cubic Curves where is called the blending function

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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

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20 Hermite Curves Since

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21 Hermite Curves so and

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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

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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

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24 Bézier Curves by definition then so

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25 Bézier Curves and

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

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

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28 Bezier Curves

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29 Convex Hull

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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

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31 B-Spline

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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

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

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

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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

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

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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

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

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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

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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

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41 Parametric Bi-Cubic Surfaces so

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42 Hermite Surfaces

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43 Bézier Surfaces

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44 B-Spline Surfaces

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45 Normals to Surfaces

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46 Quadric Surfaces implicit surface equation an alternative representation

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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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