10.4.1 Linear approximation

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10.4.1 Linear approximation

Linear approximation is the simplest and most often provides a good initial approximation, since it is a solution to the system of geodesic equations (10.17) to (10.20) when we neglect all the nonlinear terms in the right hand side. We connect the two end points in the parameter space by a straight line and define a uniform mesh or grid by a set of $k=1,2,\ldots,m$ points as shown in Fig. 10.2 (a). Therefore we have

$\displaystyle u_k = u_A + \frac{u_B - u_A}{m - 1}(k-1)\;,$			(10.56)
$\displaystyle v_k = v_A + \frac{v_B - v_A}{m - 1}(k-1)\;.$			(10.57)

When the uniform mesh in the parameter space is mapped onto the surface, the corresponding arc length mesh will not be in general uniform.

**Figure 10.2:** Initial approximations (adapted from [247]): (a) linear approximation, (b) circular arc approximation
$\begin{figure}\centerline{ \psfig{figure=fig/init.PS,height=3.0in} } \vspace*{-10mm}\end{figure}$

If we assume that $u_A \neq u_B$ then

$\displaystyle \frac{dv}{du}= \frac{v_B - v_A}{u_B - u_A} \equiv \phi\;,$

(10.58)

hence

$\displaystyle \frac{dv}{ds}= \phi\frac{du}{ds}\;.$

(10.59)

By substituting this relation into the first fundamental form we obtain

$\displaystyle E\left(\frac{du}{ds}\right)^2 + 2F\phi\left(\frac{du}{ds}\right)^2+ G \phi^2\left(\frac{du}{ds}\right)^2 = 1\;.$

(10.60)

Thus

$\displaystyle p_k = \frac{du}{ds} = \pm \frac{1}{\sqrt{E_k + 2F_k\phi + G_k\phi^2}}\;,$			(10.61)
$\displaystyle q_k = \frac{dv}{ds} = \pm \frac{\phi}{\sqrt{E_k + 2F_k\phi + G_k\phi^2}}\;.$			(10.62)

When , it is easy to find that and $q_k= \frac{1}{\sqrt{G_k}}$ . It is well known that conjugate points do not exist on regions of a surface where the Gaussian curvature is negative [412]. Therefore, the linear approximation will typically provide a good initial approximation to the geodesic path in those regions.

Next: 10.4.2 Circular arc approximation Up: 10.4 Initial approximation Previous: 10.4 Initial approximation Contents Index

December 2009