Conversion from Spherical Coordinates to Cartesian Coordinates

Theorem

Let $S$ be ordinary $3$-space.

Let a Cartesian $3$-space $\CC$ be applied to $S$.

Let a spherical coordinate system $\PP$ be superimposed upon $\CC$ such that:

$(1): \quad$ The origin of $\CC$ coincides with the pole of $\PP$.

$(2): \quad$ The $x$-axis of $\CC$ coincides with the horizontal axis of $\PP$.

$(3): \quad$ The $z$-axis of $\CC$ coincides with the polar axis of $\PP$.

Let $p$ be a point in $S$.

Let $p$ be specified as $p = \polar {r, \theta, \phi}$ expressed in the spherical coordinates of $\PP$.

Then $p$ is expressed as $\tuple {r \sin \theta \cos \phi, r \sin \theta \sin \phi, r \cos \theta}$ in $\CC$.

Contrariwise, let $p$ be expressed as $\tuple {x, y, z}$ in the cartesian coordinates of $\CC$.

Then $p$ is expressed in $\PP$ as:

$p = \polar {\sqrt {x^2 + y^2 + z^2}, \arctan \dfrac {\sqrt x^2 + y^2} z, \arctan \dfrac y x}$

where:

$\theta$ is such that $0 \le \theta < \pi$

$\phi$ is such that $x : y : r \sin \theta = \cos \phi : \sin \phi : 1$

$\arctan$ denotes the arctangent function.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): spherical coordinate system
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): spherical coordinate system