Skip to main content

Questions on spherical coordinates, a three-dimensional coordinate system where a point is represented in terms of its distance from the origin, and its latitude and longitude angles (or complements thereof).

To convert between Cartesian and spherical coordinates one can use the transformations $$ \begin{cases} x= \rho \cos(\theta)\sin(\phi) \\ y= \rho \sin(\theta)\sin(\phi) \\ z= \rho \cos(\phi) \end{cases}; $$ $$ \begin{cases} \rho = \sqrt{x^2+y^2+z^2} \\ \theta = \text{atan2}(y/x) \\ \phi = \arccos(z/r) \end{cases}, $$where $\text{atan2}$ accounts for the signs of $x,y$. There is some ambiguity regarding the naming of the angles $\theta,\phi$; $r$ is often used instead of $\rho$ as well.

The Jacobian determinant is $\rho^2 \sin(\phi)$.