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lecnotes8_fixed

# lecnotes8_fixed - 13.472J/1.128J/2.158J/16.940J...

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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

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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 ezier curve that interpolates P 1 and P N (end points). Solution 1. Parametrization by chord-length method Let ˆ u 1 = 0; ˆ ˆ 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 ˆ u i = u i /d (8.3) which is normalized as u i [0 , 1] with u 1 = 0 and u N = 1. 2. Linear equations A cubic ezier curve is defined as 3 Q ( u ) = Q i B i, 3 ( u ) , 0 u 1 (8.4) i =0 where B i, 3 ( u ) are the cubic Bernstein polynomials. 2
Obviously, the boundary conditions require Q 0 = 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 0 , 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 θE 2 = 0 B T Bq B T l = 0 (8.9) θ 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. Notes: 1. The choice of internal knots in the B-spline basis should reﬂect any knowledge of deriva- tive discontinuity in the data, as shown in Figure 8.1. 2. Greater density of knots is needed in rapidly changing parts of the shape. 3. NAG routines for approximate fitting of cubic B-splines [9] (a) Curves: E02BAF (b) Surfaces: E02DAF & E02ZAF 4. NAG routines on least square problems provide more ﬂexibility. 3

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Double knot for cubics Figure 8.1: Set of data reﬂecting a possible discontinuity of tangent vector. 8.2 Fairing of Curves and Surfaces 8.2.1 Properties and Definition Motivation: 1. Spline curves resulting from (a) interpolation of points; (b) manipulation of polygon, usually need fairing.
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