  The direct geodesic problem on an oblate spheroid is described as an initial
value problem and is solved numerically in geodetic and Cartesian coordinates.
The geodesic equations are formulated by means of the theory of differential
geometry. The initial value problem under consideration is reduced to a system
of first-order ordinary differential equations, which is solved using a
numerical method. The solution provides the coordinates and the azimuths at any
point along the geodesic. The Clairaut constant is not assumed known but it is
computed, allowing to check the precision of the method. An extended data set
of geodesics is used, in order to evaluate the performance of the method in
each coordinate system. The results for the direct geodesic problem are
validated by comparison to Karney's method. We conclude that a complete,
stable, precise, accurate and fast solution of the problem in Cartesian
coordinates is accomplished.
