Figure 10.10 shows a planar cubic Bézier curve and point
(0.3, 0.2) in the
parameter domain, which will be mapped onto
the wave-like B-spline surface as shown in Fig.
10.11. The algorithm finds three geodesic paths
,
and
, whose tangent vectors at
,
and
are orthogonal to the tangent vectors at the curve at those
points. Table 10.3 shows the list of computational
conditions and results. The entries
,
and
are the
parameter values of the curve corresponding to the first two initial
approximations for the secant method and the solution value. The
following entries,
,
and
are the number of mesh points,
correction factors for the Newton and secant methods. Tolerances for
the convergence of Newton and secant methods are given by
,
. The shortest path is given by path
with
=0.275.
Table 10.3:
Numerical conditions and results for the computation of the geodesic path
between a point and a curve on wave-like surface (adapted from [247])
Iter.
Geodesic distance
0
0.02
0.266
101
0.2
0.05
1.0E-3
1.0E-6
16
0.275
1
0.98
0.727
101
0.2
0.05
1.0E-3
1.0E-6
14
0.371
0.496
0.516
0.579
101
0.2
0.05
1.0E-3
1.0E-6
8
0.387
Figure 10.10:
Cubic
Bézier curve in the parameter domain (adapted from [247])
Figure 10.11:
Geodesic
paths from point
to Bézier curve on the wave-like bicubic B-spline surface (adapted from [247])