I am attempting to implement this, pages 11/12 method to plot geodesic equations on the surface of an object, beginning with the sphere. I would like to be able to guarantee that the geodetic length of my drawn geodesics are constant for all initial conditions after one time step.
More specifically, the method describes a way to solve the geodesic equations given an initial point and a velocity at that point in local coordinates. I have found experimentally that depending on the initial point and velocity, the length of the drawn geodesic changes. I would like to know if there is a way to guarantee that the geodesics are of constant length, regardless of the starting point, by choosing my initial velocities in such a way. Alternatively, can explain perhaps explain how different start points and velocities would change the length of a geodesic over one time step? Thanks!
A curve that satisfies those differential equations must have constant speed (but need not have speed $1$). Thus, to get the same length geodesics for a given time step, you must choose the initial speeds to be the same. You compute the square of the speed by taking $E(u')^2 + G(v')^2$, so make sure these quantities are always the same. The starting points should not matter; of course, when you compute the initial speed, you need the values of $E$ and $G$ at your starting point.