next up previous contents index
Next: 10.3 Two point boundary Up: 10.2 Geodesic equation Previous: 10.2.1 Parametric surfaces   Contents   Index

10.2.2 Implicit surfaces

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 $ {\bf r}={\bf r}(s)$ or $ x=x(s)$ , $ y=y(s)$ , $ z=z(s)$ on an implicit surface $ f(x,y,z)=0$ . By substituting $ {\bf t}=(x', y', z')^T$ , $ \frac{d{\bf t}}{ds}=(x'', y'', z'')^T$ , $ {\bf N}=\frac{\nabla f}{\vert\nabla f\vert}$ , into (10.4), we obtain the expression for the geodesic curvature of a curve on the implicit surface
$\displaystyle \kappa_g=\frac{(y'z'' - z'y'')f_x + (z'x'' - x'z'')f_y + (x'y'' - y'x'')f_z}{\sqrt{f_x^2 + f_y^2 + f_z^2}}\;.$     (10.25)

For the sake of completeness, the geodesic curvature for a non-arc-length parametrized curve is given by
$\displaystyle \kappa_g = \frac{(\dot{y}\ddot{z} - \dot{z}\ddot{y})f_x + (\dot{z... ...dot{x}^2 + \dot{y}^2 + \dot{z}^2)^{\frac{3}{2}}\sqrt{f_x^2 + f_y^2 + f_z^2}}\;.$     (10.26)

Now if we set $ \kappa_g=0$ , we deduce

$\displaystyle (y'z'' - z'y'')f_x + (z'x'' - x'z'')f_y + (x'y'' - y'x'')f_z = 0\;.$     (10.27)

Since the unit tangent vector $ (x', y', z')$ and the curvature vector $ (x'',y'',z'')$ of the geodesic curve are orthogonal to each other, we have
$\displaystyle x'x''+y'y''+z'z''=0\;.$     (10.28)

The third equation can be derived from (6.21)
$\displaystyle f_{xx} (x')^2 + f_{yy} (y')^2 + f_{zz} (z')^2 +2(f_{xy}x'y' + f_{yz}y'z' + f_{xz}x'z') + f_xx'' + f_yy'' + f_zz''=0\;.$     (10.29)

Now we solve the linear system of three equations (10.27) to (10.29) in $ (x'',y'',z'')$ , assuming that $ (z'f_y - y'f_z)^2 + (x'f_z - z'f_x)^2 + (y'f_x - x'f_y)^2$ does not vanish, yielding
$\displaystyle x''= \frac{(x'f_z - z'f_x)z' + (x'f_y - y'f_x)y'}{(z'f_y - y'f_z)^2 + (x'f_z - z'f_x)^2 + (y'f_x - x'f_y)^2}\Lambda\;,$     (10.30)
$\displaystyle y''= \frac{(y'f_z - z'f_y)z' + (y'f_x - x'f_y)x'}{(z'f_y - y'f_z)^2 + (x'f_z - z'f_x)^2 + (y'f_x - x'f_y)^2}\Lambda\;,$     (10.31)
$\displaystyle z''= \frac{(z'f_y - y'f_z)y' + (z'f_x - x'f_z)x'}{(z'f_y - y'f_z)^2 + (x'f_z - z'f_x)^2 + (y'f_x - x'f_y)^2}\Lambda\;,$     (10.32)

where $ \Lambda = f_{xx} (x')^2 + f_{yy} (y')^2 + f_{zz} (z')^2 +2(f_{xy}x'y' + f_{yz}y'z' + f_{xz}x'z')$ . These three second order differential equations can be rewritten as a system of six first order differential equations:
$\displaystyle x'= p\;,$     (10.33)
$\displaystyle y'= q\;,$     (10.34)
$\displaystyle z'= r\;,$     (10.35)
$\displaystyle p'= \frac{(pf_z - rf_x)r + (pf_y - qf_x)q}{(rf_y - qf_z)^2 + (pf_z - rf_x)^2 + (qf_x - pf_y)^2}\Lambda\;,$     (10.36)
$\displaystyle q'= \frac{(qf_z - rf_y)r + (qf_x - pf_y)p}{(rf_y - qf_z)^2 + (pf_z - rf_x)^2 + (qf_x - pf_y)^2}\Lambda\;,$     (10.37)
$\displaystyle r'= \frac{(rf_y - qf_z)q + (rf_x - pf_z)p}{(rf_y - qf_z)^2 + (pf_z - rf_x)^2 + (qf_x - pf_y)^2}\Lambda\;.$     (10.38)

Figure 10.1 shows a geodesic on an ellipsoid ( $ \frac{x^2}{9}+ \frac{y^2}{4} + z^2=1$ ) computed by integrating the above system of six first order differential equations as an initial value problem. The initial values are given by $ (x, y, z)$ = (0, 2, 0), $ (p,q,r)$ = $ \left( \frac{\sqrt{2}}{2}, 0, \frac{\sqrt{2}}{2}\right)$ and the integration is terminated at $ (x, y, z)$ =(2.439, -0.726, 0.456).

Figure 10.1: Geodesics on an ellipsoid
\begin{figure}\centerline{ \psfig{file=fig/geo_imp.eps,height=3.0in} }\end{figure}

next up previous contents index
Next: 10.3 Two point boundary Up: 10.2 Geodesic equation Previous: 10.2.1 Parametric surfaces   Contents   Index
December 2009