# Geodesic equations in polar coordinates (Euclidean space)

I tried to derive the geodesics in polar coordinate system (which should be a straight line since the metric is still Euclidean), and arrived the same equations as in this question: How to calculate the geodesics in polar coordinates? I assumed that all parametric equations of straight lines should satisfy these equations, however it seems not to be the case. For instance, $$(r, \theta) = (a \sec t, t)$$ does not seem to satisfy the geodesic equations. Why is it so?

• The image of your curve is a geodesic. But it does not satisfy the geodesic equations because it is not parametrized proportional to arclength. – lEm Mar 23 at 13:16