next up previous contents index
Next: 10.7 Geodesic offsets Up: 10.6 Numerical applications Previous: 10.6.1 Geodesic path between   Contents   Index

10.6.2 Geodesic path between a point and a curve

Figure 10.10 shows a planar cubic Bézier curve and point $ A$ (0.3, 0.2) in the $ uv$ 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 $ AB$ , $ AB'$ and $ AB''$ , whose tangent vectors at $ B$ , $ B'$ and $ B''$ 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 $ t_1$ , $ t_2$ and $ t_B$ 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, $ m$ , $ \mu$ and $ \nu$ 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 $ \varepsilon_N$ , $ \varepsilon_S$ . The shortest path is given by path $ AB$ with $ s$ =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])
$ t_1$ $ t_2$ $ t_B$ $ m$ $ \mu$ $ \nu$ $ \varepsilon_N$ $ \varepsilon_S$ 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])
\begin{figure}\vspace*{15mm} \centerline{ \psfig{figure=fig/cubic2D.PS,height=4.0in} } \vspace*{-10mm} \end{figure}

Figure 10.11: Geodesic paths from point $ A$ to Bézier curve on the wave-like bicubic B-spline surface (adapted from [247])
\begin{figure}\centerline{ \psfig{figure=fig/cubic_cwave16_0302.PS,height=4.5in} } \end{figure}

next up previous contents index
Next: 10.7 Geodesic offsets Up: 10.6 Numerical applications Previous: 10.6.1 Geodesic path between   Contents   Index
December 2009