Spherical polar coordinates

Spherical polar coordinates

In mathematics and physics, spherical polar coordinates form a coordinate system for the three-dimensional real space $\scriptstyle \mathbb {R} ^{3}$. Three numbers, two angles and a length specify any point in $\scriptstyle \mathbb {R} ^{3}$. Sperical polar coordinates are useful in cases where there is (approximate) spherical symmetry, in interactions or in boundary conditions (or in both). In such cases spherical polar coordinates often allow the separation of variables in partial differential equations and three-dimensional integrals with spherically symmetric integrands simplify drastically when transformed to spherical polar coordinates.

Definition

Let x, y, z be Cartesian coordinates of a vector $\scriptstyle {\vec {\mathbf {r} }}$ in $\scriptstyle \mathbb {R} ^{3}$, that is,

${\vec {\mathbf {r} }}=({\vec {\mathbf {e} }}_{x},\,{\vec {\mathbf {e} }}_{y},\,{\vec {\mathbf {e} }}_{z}){\begin{pmatrix}x\\y\\z\\\end{pmatrix}}\equiv x\,{\vec {\mathbf {e} }}_{x}+y\,{\vec {\mathbf {e} }}_{y}+z\,{\vec {\mathbf {e} }}_{z},$

where $\scriptstyle {\vec {\mathbf {e} }}_{x},\,{\vec {\mathbf {e} }}_{y},\,{\vec {\mathbf {e} }}_{z}$ are unit vectors along the x, y, and z axis, respectively. These axes are orthogonal and so are the unit vectors along them.

The length r of the vector $\scriptstyle {\vec {\mathbf {r} }}$ is one of the three numbers necessary to give the position of the vector in three-dimensional space. By applying the theorem of Pythagoras twice it follows that r² = x² + y² + z².

Let θ be the colatitude angle (see the figure) of the vector $\scriptstyle {\vec {\mathbf {r} }}$. In the usual geographic coordinate system used to describe a position on Earth the corresponding latitude angle has its zero at the equator, while the colatitude angle, introduced here, has its zero at the "North Pole", that is, the angle θ is zero when $\scriptstyle {\vec {\mathbf {r} }}$ is along the positive z-axis. The sum of latitude and colatitude angle of a point is 90⁰, the fact that the angles are complementary explains the name of the latter.

The angle φ gives the angle with the x-axis of the projection $\scriptstyle {\vec {\mathbf {r} }}'$ of $\scriptstyle {\vec {\mathbf {r} }}$ on the x-y plane. The angle φ is the longitude angle (also known as the azimuth angle).

Note that the projection $\scriptstyle {\vec {\mathbf {r} }}'$ has length rsinθ. The length of the projection of the latter vector on the x and y axis is therefore rsinθcosφ and rsinθsinφ, respectively. In summary, the spherical polar coordinates r, θ, and φ of $\scriptstyle {\vec {\mathbf {r} }}$ are related to its Cartesian coordinates by

${\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}$

Given a spherical polar triplet (r, θ, φ) the corresponding Cartesian coordinates are readily obtained by application of these defining equations. The figure makes clear that 0⁰ ≤ φ ≤ 360⁰, 0⁰ ≤ θ ≤ 180⁰, and r > 0.

The computation of spherical polar coordinates from a given triplet of Cartesian coordinates is somewhat more difficult due to the fact that the spherical polar coordinate system has singularities, also known as points of indeterminacy. The first such point is immediately clear: if r = 0, we have a zero vector (a point in the origin). Then θ and φ are undetermined, that is to say, any value for these two parameters will give the correct answer x = y = z = 0. Compare this to the case that one of the Cartesian coordinates is zero, say x = 0, then the other two coordinates are still determined (they fix a point in the yz-plane).

Two other points of indeterminacy are the "North" and the "South Pole", θ = 0⁰ and θ = 180⁰, respectively (while r ≠ 0). On both poles the longitudinal angle φ is undetermined.

So, when going from Cartesian coordinates to spherical polar coordinates, one has to watch for the singularities, especially when the transformation is performed by a computer program. The steps are

${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r),\quad r\neq 0\\r'&=r\sin \theta ={\sqrt {x^{2}+y^{2}}}\\\phi &={\begin{cases}\arccos(x/r')&\quad {\hbox{if}}\quad y\geq 0,\quad r'\neq 0\\360^{0}-\arccos(x/r')&\quad {\hbox{if}}\quad y<0,\quad r'\neq 0\\\end{cases}}\end{aligned}}$

(To be continued)