Chapter7

# Chapter7 - Representation and Manipulation of Surfaces Ch 7...

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Representation and Manipulation of Surfaces – Ch. 7 Objectives: Types of surface equations. Bilinear surface. Coon’s patch and bicubic patch. Bezier and B-spline surfaces. Nonuniform rational B-spline (NURBS) surfaces. Interpolation surfaces and Intersection of surfaces.

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Just as with curves, surface equations classified as either parametric or nonparametric equations. Types of Surface Equations
Types of Curve Equations - Example Y X u Z Nonparametric equation of a sphere: Parametric equation of a sphere: ( 29 ( 29 2 2 , 2 0 sin cos sin cos cos , π - + + = v u v R v u R v u R v u k j i P 2 2 2 2 2 2 2 or 0 y x R z R z y x - - ± = = - + +

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Bilinear Surface Figure 7.1 A bilinear surface and its data points. Derived by interpolating four data points using linear equations in parameters u and v so resulting surface has four points at its corners. Ratio of line segments: v v v v v v v v - = - - - = - - 1 , 1 , 1 1 , 1 0 , 1 , 1 , 0 1 , 0 0 , 0 , 0 P P P P P P P P ( 29 ( 29 1 , 1 0 , 1 , 1 1 , 0 0 , 0 , 0 1 1 P P P P P P v v v v v v + - = + - = Solving gives: Similarly obtain P ( u , v ): ( 29 ( 29 v v u u v u , 1 , 0 1 , P P P + - =
•Solve for P ( u , v ) by substituting for P 0, v and P 1, v : ( 29 ( 29 ( 29 [ ] ( 29 [ ] ( 29 ( 29 ( 29 ( 29 [ ] ( 29 1 0 , 1 0 1 1 1 1 1 1 1 , 1 , 1 1 , 0 0 , 1 0 , 0 1 , 1 0 , 1 1 , 0 0 , 0 - - - - = + - + + - - = v u uv v u v u v u v v u v v u v u P P P P P P P P P Substituting appropriate combinations into u and v of 0 and 1 verifies data points at corners. Bilinear surface also obtained by simply blending the effects of four corner points after they are weighted by blending functions: Bilinear Surface ( 29 ( 29 ( 29 ( 29 [ ] uv v u v u v u - - - - 1 1 1 1

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Coon’s Patch Figure 7.2 Four boundary curves defining a Coon’s patch. Four boundary curves blended to form surface. Patch is surface segment corresponding to parameter region 0 ≤ u ≤ 1, 0 ≤ v ≤ 1. Note direction of u and v intervals. ( 29 ( 29 ( 29 ( 29 v u v u v u 1 0 1 1 , P P P + - = P 2 ( u , v ) Interpolation of curves in u direction to define surface (note – not bounded by Q 1 ( u ) and Q 2 ( u )): ( 29 ( 29 ( 29 ( 29 u v u v v u 1 0 2 1 , Q Q P + - = Interpolation of curves in v direction to define surface (note – not bounded by P 1 ( u ) and P 2 ( u )):
•Define surface P 3 ( u , v ) by adding P 1 ( u , v ) and P 2 ( u , v ): Substituting limit values of u and v into above: Coon’s Patch ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 u v u v v u v u v u 1 0 1 0 3 1 1 , Q Q P P P + - + + - = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 1 , 0 0 1 0 , 1 1 1 , 1 0 0 1 , 0 1 0 1 3 1 0 0 3 1 0 1 3 1 0 0 3 P P Q P P P Q P Q Q P P Q Q P P u u u u u u u u v v v v v v v v + - + = + - + = + - + = + - + =

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Underlined terms linear interpolation between end points of corresponding boundary curve. •They drop out if P 3 ( u , v ) satisfies boundary curves. Therefore, correct equation of Coon’s patch found by subtracting bilinear surface equation from P 3 ( u , v ): Coon’s Patch ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 , 1 1 , 0 0 , 1 0 , 0 1 0 1 0 1 1 1 1 1 1 , P P P P Q Q P P P uv v u v u v u u v u v v u v u v u - - - - - - - - + - + + - = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 or 1 by given 0 or 1 by given 1 or 0 by given 0 or 0 by given 1 1 1 , 1 1 0 1 , 0 0 1 0 , 1 0 0 0 , 0 Q P P Q P P Q P P Q P P
Surface is represented by polynomial equation with degree 3 in parameters u and v : Bicubic Patch ( 29 ( 29 1 0 , 1 0 , 3 0 3 0 = ∑∑ = = v u v u v u i j j i ij a

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Chapter7 - Representation and Manipulation of Surfaces Ch 7...

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