$x=\frac{\sinh \left(v\right)}{\cosh \left(v\right)-\cos \left(u\right)}$

$y=\frac{\sin \left(u\right)}{\cosh \left(v\right)-\cos \left(u\right)}$

$x=\frac{u^2-v^2}{2{\left(u^2+v^2\right)}^2}$

$y=\frac{uv}{{\left(u^2+v^2\right)}^2}$

$x=u$

$y=v$

$x=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}+{\ⅇ}^u\cos \left(v\right)+1}}{2}$

$y=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}-{\ⅇ}^u\cos \left(v\right)-1}}{2}$$$

$x=\cosh \left(u\right)\cos \left(v\right)$

$y=\sinh \left(u\right)\sin \left(v\right)$

$x=\sqrt{\sqrt{u^2+v^2}+u}$

$y=\sqrt{\sqrt{u^2+v^2}-u}$

$x=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}+{\ⅇ}^u\cos \left(v\right)+1}}{2\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}}$

$y=\frac{a\sqrt{2}\sqrt{\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}-{\ⅇ}^u\cos \left(v\right)-1}}{2\sqrt{{\ⅇ}^{2u}+2{\ⅇ}^u\cos \left(v\right)+1}}$

$x=\frac{a\cosh \left(u\right)\cos \left(v\right)}{{\cosh \left(u\right)}^2-{\sin \left(v\right)}^2}$

$y=\frac{a\sinh \left(u\right)\sin \left(v\right)}{{\cosh \left(u\right)}^2-{\sin \left(v\right)}^2}$

$x=\frac{a\ln \left(u^2+v^2\right)}{\mathrm{\pi }}$

$y=\frac{2a\arctan \left(\frac{v}{u}\right)}{\mathrm{\pi }}$

$x=\frac{a\ln \left({\cosh \left(u\right)}^2-{\sin \left(v\right)}^2\right)}{\mathrm{\pi }}$

$y=\frac{2a\arctan \left(\tanh \left(u\right)\tan \left(v\right)\right)}{\mathrm{\pi }}$

$x=\frac{a\left(u+1+{\ⅇ}^u\cos \left(v\right)\right)}{\mathrm{\pi }}$

$y=\frac{a\left(v+{\ⅇ}^u\sin \left(v\right)\right)}{\mathrm{\pi }}$

$x=\frac{u^2}{2}-\frac{v^2}{2}$

$y=uv$

$x=u\cos \left(v\right)$

$y=u\sin \left(v\right)$

$x=\frac{\sqrt{\sqrt{u^2+v^2}+u}}{\sqrt{u^2+v^2}}$

$y=\frac{\sqrt{\sqrt{u^2+v^2}-u}}{\sqrt{u^2+v^2}}$

$x=\frac{u}{u^2+v^2}$

$y=\frac{v}{u^2+v^2}$