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# lecnotes9_fixed - 13.472J/1.128J/2.158J/16.940J...

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13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 9 N. M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright c 2003 Massachusetts Institute of Technology Contents 9 Blending Surfaces 2 9.1 Examples and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9.2 Blending surface approximation in terms of B-splines . . . . . . . . . . . . . . 4 9.2.1 Blend construction through a procedural “lofted” surface . . . . . . . . 5 9.3 Spherical and circular blending in terms of generalized cylinders . . . . . . . . 7 9.4 Blending of implicit algebraic surfaces . . . . . . . . . . . . . . . . . . . . . . . 11 9.5 Blending as a boundary value problem . . . . . . . . . . . . . . . . . . . . . . 12 9.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 9.5.2 Example: 2 nd order (Laplace) equation . . . . . . . . . . . . . . . . . . 13 9.5.3 Mapping – boundary value problem . . . . . . . . . . . . . . . . . . . . 17 9.5.4 Position and tangent plane continuity . . . . . . . . . . . . . . . . . . . 19 9.5.5 Curvature continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 9.5.6 Multisided blending surfaces . . . . . . . . . . . . . . . . . . . . . . . . 21 Bibliography 24 1

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Lecture 9 Blending Surfaces 9.1 Examples and motivation Blending surfaces, providing a smooth connection between various primary or functional sur- faces, are very common in CAD. Examples include blending surfaces between: Fuselage and wings of airplanes Propeller or turbine blade and hub Bulbous bow and ship hull Primary faces of solid models. Blending (or filleting) surfaces are also byproducts of manufacturing processes such as NC milling with a ball or disk cutter. As a result of continuity conditions, blending surfaces are of higher order, or involve a more complex formulation than the underlying surfaces to be joined. For a detailed review, see Woodwark [15], and Hoschek and Lasser [9] (chapter 14). 2
Example Join bicubic patches along arbitrary cubic linkage curves in parametric space (see Figure 9.1). R ( u , v ) 3 3 Q ( w , s ) 3 3 v = v ( t ) 3 u = u ( t ) 3 w = w ( t ) 3 s = s ( t ) 3 Figure 9.1: Bicubic patches joined along arbitrary cubic linkage curves. Linkage curve 1 is: R 1 ( t ) R ( t ) = R ( u 3 , v 3 ) R ( t 18 ) , if u = u ( t 3 ) , v = v ( t 3 ) (9.1) Similarly, curve 2 is: R 2 ( t ) Q ( t ) = Q ( w 3 , s 3 ) Q ( t 18 ) , if w = w ( t 3 ) , s = s ( t 3 ) (9.2) So position continuity alone requires a high degree surface in the t parameter direction (of degree 18 in this example). High degree surfaces are expensive to evaluate (e.g. the de Casteljau or Cox-de Boor algorithms have quadratic complexity in the degree of the curve or surface), may lead to greater inaccuracy of evaluation (as the degree increases), and are difficult to process in a solid modeling environment (e.g. through intersections). Consequently, researchers have developed: Approximations of blending surfaces with low order B-spline surfaces Procedural definitions of blending surfaces (e.g. “lofted” surfaces, generalized cylinders) in order to reduce some of these problems. 3

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9.2 Blending surface approximation in terms of B-splines Let us consider two parametric surface patches r m 1 n 1 ( u, v ) , r m 2 n 2 ( x, y ) and linkage curves R 1 ( t ) = [ u ( t ) , v ( t )] , R 2 ( t ) = [ x ( t ) , y ( t )] defined in the parameter spaces of the two patches, respectively.
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