The Geodesic Equation

What does it mean to be straight? That the direction of the line does not change. In other words, the unit tangent vector to the curve remains constant. Note that the speed does not need to be constant; only the direction remains the same. However, if we are only interested in the shape of the line, and not in the particular manner in which we move along it, it is convenient to use a description in which the velocity itself is constant. One way to do this is to use arclength itself as the parameter along the curve; such parameterizations are also called unit speed. In two dimensions, then, we can describe any curve by its position (x, y), with the coordinates x and y being thought of as functions of arclength s