= Normalization of the Poincare disc metric
{title2=$ds^2=4|dz|^2/(1-|z|^2)^2$}
Transferring the upper half-plane <hyperbolic metric> $|dw|^2/(\operatorname{Im}w)^2$ through the <Cayley transform> gives $ds^2=4|dz|^2/(1-|z|^2)^2$ on the unit disc. Radial integration gives distance $2\operatorname{artanh}r$ from zero to $r$. The alternative convention $d\widetilde s^2=|dz|^2/(1-|z|^2)^2$ divides the metric tensor by four and every length by two, giving $\operatorname{artanh}r$. Both have the same <isometries> and unparametrized <geodesics>, but their <Gaussian curvatures> are respectively minus one and minus four. In general scaling a <Riemannian metric> by $c>0$ scales lengths by $\sqrt c$ and <curvature> by $1/c$.
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