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

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

$z=w$

$x=\frac{\sin \left(u\right)\cos \left(w\right)}{d}$

$y=\frac{\sin \left(u\right)\sin \left(w\right)}{d}$

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

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

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

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

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

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

$z=w$

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

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

$z=w$

$x=\sqrt{\frac{\left(a^2-u\right)\left(a^2-v\right)\left(a^2-w\right)}{\left(a^2-b^2\right)\left(a^2-c^2\right)}}$

$y=\sqrt{\frac{\left(b^2-u\right)\left(b^2-v\right)\left(b^2-w\right)}{\left(-a^2+b^2\right)\left(b^2-c^2\right)}}$

$z=\sqrt{\frac{\left(c^2-u\right)\left(c^2-v\right)\left(c^2-w\right)}{\left(-a^2+c^2\right)\left(-b^2+c^2\right)}}$

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

$y=\sqrt{\frac{\left(b^2-u\right)\left(b^2-v\right)\left(b^2-w\right)}{-a^2+b^2}}$

$z=\frac{a^2}{2}+\frac{b^2}{2}-\frac{u}{2}-\frac{v}{2}-\frac{w}{2}$

$x=\frac{uvw}{ab}$

$y=\frac{u\sqrt{\frac{\left(-b^2+v^2\right)\left(b^2-w^2\right)}{a^2-b^2}}}{b}$

$z=\frac{u\sqrt{\frac{\left(a^2-v^2\right)\left(a^2-w^2\right)}{a^2-b^2}}}{a}$

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

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

$z=w$

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

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

$z=w$

$x=\frac{uvw}{ab}$

$y=\frac{\sqrt{\frac{\left(-b^2+u^2\right)\left(-b^2+v^2\right)\left(b^2-w^2\right)}{a^2-b^2}}}{b}$

$z=\frac{\sqrt{\frac{\left(-a^2+u^2\right)\left(a^2-v^2\right)\left(a^2-w^2\right)}{a^2-b^2}}}{a}$

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

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

$z=w$

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

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

$z=w$

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

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

$z=w$

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

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

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

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

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

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

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

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

$z=w$

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

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

$z=w$

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

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

$z=w$

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

$y=a\cosh \left(u\right)\sin \left(v\right)\sin \left(w\right)$

$z=a\sinh \left(u\right)\cos \left(v\right)$

$x=uv\cos \left(w\right)$

$y=uv\sin \left(w\right)$

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

$x=2\sqrt{\frac{\left(u-a\right)\left(a-v\right)\left(a-w\right)}{a-b}}$

$y=2\sqrt{\frac{\left(u-b\right)\left(b-v\right)\left(b-w\right)}{a-b}}$

$z=u+v+w-a-b$

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

$y=uv$

$z=w$

$x=a\sinh \left(u\right)\sin \left(v\right)\cos \left(w\right)$

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

$z=a\cosh \left(u\right)\cos \left(v\right)$

$x=u$

$y=v$

$z=w$

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

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

$z=w$

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

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

$z=\frac{w}{u^2+v^2+w^2}$

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

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

$z=u\cos \left(w\right)$

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

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

$z=w$

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

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

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

$x=\frac{a\sinh \left(v\right)\cos \left(w\right)}{d}$

$y=\frac{a\sinh \left(v\right)\sin \left(w\right)}{d}$

$z=\frac{a\sin \left(u\right)}{d}$  where $d=\cosh \left(v\right)-\cos \left(u\right)$