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Ok, we have the metric $ds^2=\frac{1}{y^2}$ defined in the upper half plane $U=\{(x,y)\in\mathbb{R}^2|y>0\}$.

I know two geodesics are $x(t)=a-b\cos{t}$ and $y(t)=b\sin{t}$. What are some others? And why would they be geodesics of this metric? (Particularly this second question is important.)

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up vote 3 down vote accepted

This is the hyperbolic metric in the upper-half plane and we know that the geodesic of this model are the half-circles orthogonal to the $x$-axis and the vertical lines

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