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Unformatted text preview: 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 8 N. M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA c Copyright 2003 Massachusetts Institute of Technology Contents 8 Fitting, Fairing and Generalized Cylinders 2 8.1 Least Squares Method of Curve Fitting . . . . . . . . . . . . . . . . . . . . . . 2 8.2 Fairing of Curves and Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8.2.1 Properties and Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 4 8.2.2 Curve Interrogation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8.2.3 Fairing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8.2.4 Surface Fairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 8.3 Generalized Cylinders: Motivation and Definitions . . . . . . . . . . . . . . . . 12 8.3.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 8.3.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 8.4 Degeneracies of Generalized Cylinders . . . . . . . . . . . . . . . . . . . . . . . 16 8.5 Properties of Generalized Cylinders . . . . . . . . . . . . . . . . . . . . . . . . 20 8.6 Discrete Generalized Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Bibliography 22 Reading in the Textbook Chapter 11, pp. 353- 365 1 Lecture 8 Fitting, Fairing and Generalized Cylinders 8.1 Least Squares Method of Curve Fitting Example problem Given N points P i , i = 1 , 2 , ..., N ( N 4), construct an approximating cubic B ezier curve that interpolates P 1 and P N (end points). Solution 1. Parametrization by chord-length method Let u 1 = 0; u i +1 = u i + d i +1 , i = 1 , 2 , ..., N 1 (8.1) where d i +1 = | P i +1 P i | is the chord length between two consecutive points. The overall chord length is N d = d i (8.2) i =2 The parametric value associated with point P i u i = u i /d (8.3) which is normalized as u i [0 , 1] with u 1 = and u N = 1. 2. Linear equations A cubic B ezier curve is defined as 3 Q ( u ) = Q i B i, 3 ( u ) , u 1 (8.4) i =0 where B i, 3 ( u ) are the cubic Bernstein polynomials. 2 Obviously, the boundary conditions require Q = P 1 , Q 3 = P N . The problem is then represented as a linear system with N 2 equations and 2 unknowns: 2 Q i B i, 3 ( u j ) = P j P 1 B , 3 ( u j ) P N B 3 , 3 ( u j ) i =1 = L j , j = 2 , 3 , ..., N 1 (8.5) B or in matrix form ( N 2) 2 q 2 1 = l ( N 2) 1 (8.6) 3. Least Squares Method Define the mean square error as E 2 = | B q l | 2 (8.7) then E 2 = ( B q l ) T ( B q l ) = q T B T Bq 2 q T B T l + l T l (8.8) is a function of q and is minimized if we set E 2 = B T Bq B T l = (8.9) q B T Bq = B T l (normal equations) (8.10) q = ( B T B ) 1 B T l (formal solution) (8.11) The extension to fitting with B-splines is similarly formulated....
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