We know that the parametric equation of circles of radius r and center at origin is given by $\gamma (t) = (r\cos t, r \sin t)$ and any arbitrary closed curve can be written in the form $\gamma (t) = (r(t)\cos t, r(t) \sin t)$. Can someone help me please to write parametric equation of circle or any other closed curve in hyperbolic plane? I was thinking about $\gamma (t) = (r(t)\cosh t, r(t) \sinh t)$, but it will not work as it has different radius than the Euclidean circle. and also origin cannot be the centre
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This depends on the coordinate system you are working in. Are you using the Poincaré upper half-plane model? The Poincaré ball model? The Klein model? A different model? Please specify. – Magma Dec 28 '21 at 18:43
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It's hyperbolic plane. Means it's the upper half plane and with the metric $\frac{dx^2+dy^2}{y^2}$ – ranadip ganguly Dec 29 '21 at 18:08