10.6.2 Geodesic path between a point and a curve

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])