Unformatted text preview: Rogers, D. F.; Adams, J. A. 458 MATHEMATICAL ELEMENTS FOR. COMPUTER GRAPHICS
McGrawHill 1990 2/e If [C] is square, the deﬁning polygon net is obtained directly by matrix
inversmn, i.e., [B]=[C]‘1[D] (6—83) In this case the resulting surface passes through each data point. Although the resulting surface will be everywhere Ck‘z, 01—2 continuous, it may not be fair. Experience has shown that, in general, the fewer the, deﬁning polygon net points
the fairer the surface. I If [ C ] is not square the problem is overspeciﬁed. and a solution can only be
obtained in some mean sense. In particular the solution is given by [BJ=[[01T[01]‘1[01T[D1 (6—84) The u and w parametric values for each surface data point are obtained using
a chord length approximation (see Sec. 511). Speciﬁcally, for r data points the
parameter value at the lth data point in the u parametric direction is I
Z ng,s _ Dg—1,sl
W 9:2 “max = T
2 ngvs — Dg—ly5l
g=2 Similarly, for 5 data points in the w parametric direction U1=0 I
Z [D779 — Drrg—li w; g=2
wl = 0 = 7——
wmax
Z :lDrvg — DT,g—'1
g=2 where umax and wmax are the maximum values of the appropriate knot vectors.
Figure 6*50a shows surface data and the deﬁning polygon net generated using
this technique. Figure 6—50b shows the B—spline surface generated from this net. As previously pointed out for Bspline curve ﬁtting (see Sec. 5—11), neither
this technique, nor that of Ref. 6—5, can yield hard points or hard lines (discon~
tinuities in the ﬁrst or second derivatives) in the resulting surface. ‘ Notice from Fig. 6—50 that the deﬁning polygon net vertices obtained using
Eq. (6—84) lie anywhere in the three—dimensional plane. If the surface is to
subsequently be modiﬁed, this is inconvenient. Rogers and Fog (Ref. 6—27) have
developed a technique for iterating on the parametric values u, w that conﬁnes
the deﬁning polygon net vertices to lie in planes or along lines in three space. Figure 6—50 B—spline surface ﬁt. (a) Data points and ﬁt deﬁning polygon net; (b)
data points and generated surface. ...
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