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
points as shown in Fig.
10.2 (a). Therefore we have
(10.56)
(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
If we assume that
then
(10.58)
hence
(10.59)
By substituting this relation into the first fundamental form we
obtain
(10.60)
Thus
(10.61)
(10.62)
When
, it is easy to find that
and
. 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
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December 2009