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 Bspline 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. ...
View
Full Document
 Spring '10
 Dr.KaimingYU

Click to edit the document details