We can also derive the geodesic equation for an implicit surface by
finding an
expression of the geodesic curvature for an implicit surface. Let
us consider an arc length parametrized curve
or
,
,
on an implicit surface
. By
substituting
,
,
, into
(10.4), we obtain the expression for the geodesic curvature of
a curve on the implicit surface
(10.25)
For the sake of completeness, the geodesic curvature for a non-arc-length
parametrized curve is given by
(10.26)
Now if we set
, we deduce
(10.27)
Since the unit tangent vector
and the curvature vector
of the geodesic curve are orthogonal to each other,
we have
Now we solve the linear system of three equations
(10.27) to (10.29) in
, assuming that
does not vanish, yielding
(10.30)
(10.31)
(10.32)
where
.
These three second order differential equations can be rewritten as a system of six first order differential equations:
(10.33)
(10.34)
(10.35)
(10.36)
(10.37)
(10.38)
Figure 10.1 shows a geodesic on an ellipsoid
(
) computed by integrating the
above system of six first order differential equations as an initial
value problem. The initial values are given by
= (0, 2, 0),
=
and the integration
is terminated at
=(2.439, -0.726, 0.456).