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10.6.1 Geodesic path between two points

The first example is a wave-like bicubic B-spline surface, whose control polyhedron is a lattice of 7$ \times $ 7 vertices with uniform knot vectors in both directions and spans $ 0 \leq x \leq 1$ , $ 0 \leq y \leq 1$ . Let us compute the geodesic path between two corner points, ($ u_A$ , $ v_A$ )= (0,0) and ($ u_B$ , $ v_B$ )=(1,1). We choose $ (u_C, v_C)$ to be (0.7, 0.3) and $ (u_D, v_D)$ to be (0.3, 0.7) for the circular arc approximation. The algorithm based on circular arc approximation finds three geodesic paths, as shown in Fig. 10.5 (solid thick lines). The computational conditions such as number of mesh points, tolerance and correction factor for the Newton's method $ \varepsilon_N$ , $ \mu$ , as well as computational results such as number of iterations for convergence and the geodesic distances are listed in Table 10.1. Symbols $ Lt$ , $ Md$ , $ Rt$ refer to left, middle and right geodesic paths in Fig. 10.5. The middle geodesic path is not a minimal path ($ s$ =1.865), while the other two paths are the shortest path ($ s=1.661$ ) due to symmetry. Figure 10.6 shows how the initial approximation path (the right-most thick solid line) converges gradually to the final solution (wavy thick solid line). The intermediate paths are illustrated by the thin solid lines.


Table 10.1: Numerical conditions and results for the computation of the geodesic path between corner points of the wave-like surface (adapted from [247])
Points Tolerance Correction Iterations Geodesic distance
m $ \varepsilon_N$ factor $ \mu$ Lt Md Rt Lt Md Rt
101 1.0E-3 0.2 22 1 22 1.661 1.865 1.661



Figure 10.5: Geodesic paths on the wave-like bicubic B-spline surface between points of two corners (adapted from [247])
\begin{figure}\centerline{ \psfig{figure=fig/cwave16_0011abc.PS,height=4.5in} } \end{figure}

Figure 10.6: Convergence of the right geodesic path in Fig. 10.5 (adapted from [247])
\begin{figure}\vspace*{5mm} \centerline{ \psfig{figure=fig/cwave16_0011_conv.PS,height=4.5in} } \vspace*{-5mm}\end{figure}

The second example is a generalized cylinder represented by a biquadratic rational B-spline surface, whose control polyhedron is a lattice of 6$ \times $ 9 vertices. This surface was constructed by sweeping a circle of radius 0.5, the generatrix, along a helix ($ x=\cos t$ , $ y=-\sin t$ , $ z=\frac{t}{\pi}$ , $ 0\leq t \leq 2\pi$ ), the spine, and is approximated by rational B-spline interpolating a number of generatrices. When we keep $ u$ constant, we obtain a curve on the surface which depends only on $ v$ . This curve coincides with the generatrix. Similarly $ v=constant$ represents another iso-parametric curve which is parallel to the spine. Two end points are chosen to be $ (u_A, v_A)$ =(0, 0.4) and $ (u_B, v_B)$ =(1, 0.6) as shown in Fig. 10.7. In Fig. 10.7 three initial approximations are illustrated by the thin solid lines, while the final solutions are illustrated by the thick solid lines. The two circular arcs are determined by setting $ (u_C, v_C)$ = (0.578, 0.108), $ (u_D, v_D)$ = (0.422, 0.892). The circular arc approximation algorithm starts with these two circular arcs and converges to the two minimal geodesic paths which are shown as thick solid lines close to the initial circular arcs. The minimal geodesic paths mapped onto the generalized cylinder are depicted in Color Plate A.8 and Fig. 10.8. Then the algorithm computes the mid-points of these solutions, which is shown as a thin straight line connecting the two end points. This initial approximation converges to a sine wave-like solution. This solution is a geodesic path but it does not provide the shortest distance (see Fig. 10.9). Table 10.2 shows the list of computational conditions and results as in Table 10.1.


Table 10.2: Numerical conditions and results for the computation of the geodesic path between two points on generalized cylinder (adapted from [247])
Points Tolerance Correction Iterations Geodesic distance
$ m$ $ \varepsilon_N$ factor $ \;\mu$ Lt Md Rt Lt Md Rt
501 1.0E-2 0.2 8 10 8 5.860 6.983 5.860
1001 5.0E-3 0.2 10 10 10 5.843 6.956 5.843

Figure 10.7: Geodesic paths in the parameter domain of the generalized cylinder (adapted from [247])
\begin{figure}\vspace*{15mm} \centerline{ \psfig{figure=fig/helix2D.PS,height=4.5in} } \vspace*{-10mm}\end{figure}

Figure 10.8: Top view of the minimal geodesic paths on the generalized cylinder between two points of two circular edges (adapted from [247])
\begin{figure}\vspace*{15mm} \centerline{ \psfig{figure=fig/helix_top.PS,height=4.5in} } \vspace*{-20mm} \end{figure}

Figure 10.9: A geodesic path on the generalized cylinder which is not the shortest path (adapted from [247])
\begin{figure}\vspace*{10mm} \centerline{ \psfig{figure=fig/helix3DA.PS,height=4.5in} } \vspace*{-10mm} \end{figure}

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December 2009