Coordinate system $\left(q_1\mathrm{,}{q}_2\mathrm{,}\phi \right)\mathrm{,}\phi \in \left[\mathrm{0,2}\pi \right)$

In Cartesian coordinates $\left(x\mathrm{,}y\mathrm{,}z\right)$

$h_1$

$h_2$

$\varpi$

$\frac{h_1}{h_2\varpi }$

$\frac{h_2}{h_1\varpi }$

Spherical $\left(r\mathrm{,}\theta \mathrm{,}\phi \right)$ $r\ge \mathrm{0,}\theta \in \left[\mathrm{0,}\pi \right]$

$\{\begin{array}{l}x=r\sin \left(\theta \right)\cos \left(\phi \right)\\ y=r\sin \left(\theta \right)\sin \left(\phi \right)\\ z=r\cos \left(\theta \right)\end{array}$

1

$\frac{1}{r}$

$rsin\left(\theta \right)$

$\frac{1}{sin\left(\theta \right)}$

$\frac{1}{r^{2}sin\left(\theta \right)}$

Modified Spherical $\left(r\mathrm{,}\zeta \mathrm{,}\phi \right)$ $r\ge \mathrm{0,}\zeta \in \left[-\mathrm{1,1}\right]$

$\{\begin{array}{l}x=r\sqrt{1-{\zeta }^2}\cos \left(\phi \right)\\ y=r\sqrt{1-{\zeta }^2}\sin \left(\phi \right)\\ z=r\zeta \end{array}$

1

$\frac{\sqrt{1-{\zeta }^2}}{r}$

$r\sqrt{1-{\zeta }^2}$

$\frac{1}{1-{\zeta }^2}$

$\frac{1}{r^2}$

Prolate Spheroid $\left(\eta \mathrm{,}\theta \mathrm{,}\phi \right)$

$\eta \ge \mathrm{0,}\theta \in \left[\mathrm{0,}\pi \right]$ $\alpha >0$

$\{\begin{array}{l}x=\alpha \sinh \left(\eta \right)\sin \left(\theta \right)\cos \left(\phi \right)\\ y=\alpha \sinh \left(\eta \right)\sin \left(\theta \right)\sin \left(\phi \right)\\ z=\alpha \cosh \left(\eta \right)\cos \left(\theta \right)\end{array}$

$h_2$

$\frac{1}{a\sqrt{{\sinh }^2\left(\eta \right)+{\sin }^2\left(\theta \right)}}$

$asinh\left(\eta \right)sin\left(\theta \right)$

$\frac{h_2}{h_1\varpi }$

$\frac{1}{asinh\left(\eta \right)sin\left(\theta \right)}$

Modified Prolate Spheroid $\left(\tau \mathrm{,}\zeta \mathrm{,}\phi \right)$

$\tau \ge \mathrm{1,}\zeta \in \left[-\mathrm{1,1}\right]$ $\alpha >0$

$\{\begin{array}{l}x=\alpha \sqrt{{\tau }^2-1}\sqrt{1-{\zeta }^2}\cos \left(\phi \right)\\ y=\alpha \sqrt{{\tau }^2-1}\sqrt{1-{\zeta }^2}\sin \left(\phi \right)\\ z=\alpha \tau \zeta \end{array}$

$\frac{\sqrt{{\tau }^2-1}}{a\sqrt{{\tau }^2-{\zeta }^2}}$

$\frac{\sqrt{1-{\zeta }^2}}{a\sqrt{{\tau }^2-{\zeta }^2}}$

$a\sqrt{{\tau }^2-1}\sqrt{1-{\zeta }^2}$

$\frac{1}{a\left(1-{\zeta }^2\right)}$

$\frac{1}{a\left({\tau }^2-1\right)}$

Oblate Spheroid $\left(\eta \mathrm{,}\theta \mathrm{,}\phi \right)$

$\eta \ge \mathrm{0,}\theta \in \left[\mathrm{0,}\pi \right]$ $\alpha >0$

$\{\begin{array}{l}x=\alpha \cosh \left(\eta \right)\sin \left(\theta \right)\cos \left(\phi \right)\\ y=\alpha \cosh \left(\eta \right)\sin \left(\theta \right)\sin \left(\phi \right)\\ z=\alpha \sinh \left(\eta \right)\cos \left(\theta \right)\end{array}$

$h_2$

$\frac{1}{a\sqrt{cos{h}^2\left(\eta \right)-{\sin }^2\left(\theta \right)}}$

$acosh\left(\eta \right)sin\left(\theta \right)$

$\frac{h_2}{h_1\varpi }$

$\frac{1}{acosh\left(\eta \right)sin\left(\theta \right)}$

Modified Oblate Spheroid $\left(\lambda \mathrm{,}\zeta \mathrm{,}\phi \right)$

$\lambda \in ℝ\mathrm{,}\zeta \in \left[-\mathrm{1,1}\right]$ $\alpha >0$

$\{\begin{array}{l}x=\alpha \sqrt{{\lambda }^2+1}\sqrt{1-{\zeta }^2}\cos \left(\phi \right)\\ y=\alpha \sqrt{{\lambda }^2+1}\sqrt{1-{\zeta }^2}\sin \left(\phi \right)\\ z=\alpha \lambda \zeta \end{array}$

$\frac{\sqrt{{\lambda }^2+1}}{a\sqrt{{\lambda }^2+{\zeta }^2}}$

$\frac{\sqrt{1-{\zeta }^2}}{a\sqrt{{\lambda }^2+{\zeta }^2}}$

$a\sqrt{{\lambda }^2+1}\sqrt{1-{\zeta }^2}$

$\frac{1}{a\left(1-{\zeta }^2\right)}$

$\frac{1}{a\left(1+{\lambda }^2\right)}$

Parabolic $\left(\mu \mathrm{,}\nu \mathrm{,}\phi \right)$ $\mu \mathrm{,}\nu \ge 0$

$\{\begin{array}{l}x=\mu \nu \cos \left(\phi \right)\\ y=\mu \nu \sin \left(\phi \right)\\ z=\frac{{\mu }^2-{\nu }^2}{2}\end{array}$

$\frac{1}{\sqrt{{\mu }^2+{\nu }^2}}$