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?

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

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