Normalization of the Poincare disc metric ($d{s}^2=4|dz{|}^2/(1-|z{|}^2{)}^2$)

Transferring the upper half-plane hyperbolic metric $|dw{|}^2/(\mathrm{Im}w{)}^2$ through the Cayley transform gives $d{s}^2=4|dz{|}^2/(1-|z{|}^2{)}^2$ on the unit disc. Radial integration gives distance $2\mathrm{artanh}r$ from zero to $r$. The alternative convention $d{\tilde{s}}^2=|dz{|}^2/(1-|z{|}^2{)}^2$ divides the metric tensor by four and every length by two, giving $\mathrm{artanh}r$. 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 $1/c$.