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RE317 - 350 MATHEMATICAL ELEMENTS FOR COMPUTER GRAPHICS 0 0...

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Unformatted text preview: 350 MATHEMATICAL ELEMENTS FOR COMPUTER GRAPHICS 0 0 0.409 1.378 [B]: 3 2.874 5.591 1.377 6 0 Figure 5—56a shows the original data points, the, calculated polygon ver- tices and the resulting curve. For four polygon vertices, the knot vector with multiplicity k at the ends is ' [ 0 0 0 1 2 2 2 ] [ N ] becomes 1 0 o 0 0.15 0.662 0.188 0 [N]: 0 0.5 0.5 0 0 0.188 0.662 0.15 0 0 0 1 Multiplying by [ N ]T and taking the inverse yields 0.995 —0.21 0.106 —0.005 T _.1_ —0.21 2.684 —1.855 0.106 [NV] [NH — 0.106 —1.855 2.684 —0.21 —0.005 0.106 —0.21 0.995 Equation (5—117) then gives 0 o _ T 0.788 2.414 l31=llNlrlNll 1““ [D]: 5.212 2.414 6 0 The original data, the calculated polygon vertices and the resulting curve are shown in Fig. 5—56b. Notice that except at the ends the curve does not pass through the original data points. The above ﬁtting technique allows each of the determined deﬁning polygon points for the B—spline curve to be located anywhere in three space. In some design situations it is more useful to constrain the deﬁning polygon points to lie at a particular coordinate value, say a: =' constant. An example of such a design situation is in ﬁtting B—spline curves to existing ships’ lines. Rogers and Fog (Ref. 5—26) have developed such a technique for both curves and surfaces. Essentially, the technique iterates the parameter value of the ﬁxed coordinate until the value on the B-spline curve at the assumed parameter value calculated with the deﬁning polygons obtained using the above ﬁtting technique is within some speciﬁed amount of the ﬁxed value, i.e., lxﬁxed — xca1c| 5 error. The resulting ﬁt is less accurate but more convenient for subsequent modiﬁcation. ...
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