Classroom Tips and Techniques: Geodesics on a Surface
Robert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow
Maplesoft
Introduction
Recently, Samir Khan, one of Maplesoft's Application Engineers, posted a question from a user who asked how to obtain minimallength curves on a surface. Samir provided an interesting numeric technique for finding what is essentially a geodesic connecting two points on a surface embedded in . In this article, we implement a version of Samir's numeric approach, and also show how to obtain the differential equations governing such geodesics. We do this both with the calculus of variations (minimizing the arc length integral) and with tensor calculus techniques implemented in the DifferentialGeometry package.
Astute readers will recall that several months ago, we provided the article Tensor Calculus with the Differential Geometry Package where we found geodesics in the plane when the plane was referred to polar coordinates. In this month's article we find geodesics on a surface embedded in .
Samir Khan's Numeric Approach
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The graph of the surface defined by the function in Table 1

Table 1 Function defining a surface in

appears in Figure 1.

Figure 1 Surface described by the function in Table 1

The curve of minimal length connecting the points and is approximated by a polygonal line, that is, by a linear spline, with
segments and equispaced nodes. The coordinates of the initial and terminal nodes on this spline are given respectively by
and for the intermediate nodes we have
The coordinates at the initial and terminal points are respectively
The nodes are then
where the , are unknown. The length of the approximating spline is given by
which is simply the sum of the lengths of each segment in the spline. This length is minimized with
where the negativity constraints on the coordinates are inspired by the values at the initial and terminal points. Hence, the computed nodes are
so the resulting spline can be graphed in Figure 2.

Figure 2 Approximate geodesic on the surface in Figure 1

Minimize the Arc Length Integral
Arc length is given by the integral , where is the parameter along the curve, and . Since the radicand is necessarily nonzero, we can obtain a geodesic by minimizing the integral whose integrand is
This can be done by solving the EulerLagrange equations
which we can obtain in Maple via the computations in Table 2.
In addition to the EulerLagrange equations themselves, the EulerLagrange command from the VariationalCalculus package can provide first integrals in which the constants of integration are of the form . We first removed these first integrals, and extracted the remaining two EulerLagrange differential equations, which we solve numerically in Table 3.

Table 3 Numeric solution of the EulerLagrange differential equations

In Figure 3 where we graph the solution of the EulerLagrange equations on the surface from Figure 1, we take the precaution of adding to each coordinate along the curve so that it can be seen more clearly. The alternative would have been to make the surface sufficiently transparent that the curve, lying in the surface, would be visible.

Figure 3 Solution of the EulerLagrange equations superimposed on the surface from Figure 1

Geodesics via the Differential Geometry Packages
The GeodesicEquations command in the DifferentialGeometry package will generate the geodesic equations for the surface embedded in . The frame for this surface is established with
Now, the metric tensor (first fundamental form) for the surface must be obtained. To this end, we define the surface in radiusvector form via
then compute the tangent basis vectors as per Table 4.


Table 4 Natural tangent basis vectors on the embedded surface

The components of the covariant metric tensor are contained in the matrix
the tensor form of which is generated by
The Christoffel symbols of the second kind, namely, , are obtained with
The individual Christoffel symbols can be extracted with the commands shown in Table 5. The expressions are all lengthy, and have been suppressed.
















Table 5 The Christoffel symbols of the second kind

The geodesic equations
where and , are then generated with
and extracted with
Again, the (very lengthy) outputs have been suppressed. These two differential equations are then solved numerically as in Table 6, just as their counterparts were solved in Table 3.

Table 6 Numeric solution of geodesic equations generated by the GeodesicEquations command

Figure 4, a graph of the geodesic (raised slightly off the surface) and the surface , is obtained just as Figure 3 was.

Figure 4 Numerically computed geodesic drawn on the surface

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