Any parametrization of a geodesic is still a geodesic

Question

I am confused about the reparametrization of geodesics.

According to the definition of geodesics, a curve $\alpha$ is geodesic if it has constant speed.

Now in Euclidean spaces the geodesics and the straight line. Now what about

$\alpha(t)=(\frac{\sigma}{2}(e^{-\frac{\sigma t}{2}}-1)+x_0,y_0)$,

where $\sigma$ is constant and $(x_0,y_0)$ is a point in the plane.

From one hand as it is a line it must be geodesic, however it speed is not constant!

Any comments

Answer 1

A geodesic must have constant speed, else you have a curve whiches image is the image of a geodesic, but which is not necessarily a geodesic (but since there usually exists a reparametrization by arc length, this is not to much of a problem).