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Classroom Tips and Techniques: Geodesics on a Surface

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Classroom Tips and Techniques: Geodesics on a Surface
Robert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow



Recently, Samir Khan, one of Maplesoft's Application Engineers, posted a question from a user who asked how to obtain minimal-length 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 


Execute entire worksheet: Click "!!!" icon in the toolbar  

Execute stepwise: Click icon to the right (to initialize), then execute each command 



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 Euler-Lagrange equations 




which we can obtain in Maple via the computations in Table 2. 



Table 2   The Euler-Lagrange equations for a geodesic in a surface 


In addition to the Euler-Lagrange 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 Euler-Lagrange differential equations, which we solve numerically in Table 3. 



Table 3   Numeric solution of the Euler-Lagrange differential equations 


In Figure 3 where we graph the solution of the Euler-Lagrange 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 Euler-Lagrange 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 radius-vector 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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