Spherical coordinates in three-dimensional coordinate space

Defining spherical coordinates in three-dimensional space

Like cartesian (or rectangular) coordinates and polar coordinates, spherical coordinates are just another way to describe points in three-dimensional space.

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Rectangular coordinates are given as $(x,y,z)$.

$x$ is the distance of $(x,y,z)$ from the origin along the $x$-axis

$y$ is the distance of $(x,y,z)$ from the origin along the $y$-axis

$z$ is the distance of $(x,y,z)$ from the origin along the $z$-axis

Spherical coordinates are given as $(\rho,\theta,\phi)$

$\rho$ is the distance of $(\rho,\theta,\phi)$ from the origin, $\rho\geq 0$

$\theta$ is the angle between $r$ (the shadow of the line connecting $(\rho,\theta,\phi)$ to the origin) and the positive direction of the $x$-axis

$\phi$ is the angle between the line connecting $(\rho,\theta,\phi)$ to the origin and the positive direction of the $z$-axis, $0\leq\phi\leq\pi$

To convert between spherical coordinates and rectangular coordinates, we will need to use the formulas

$x=\rho\sin{\phi}\cos{\theta}$

$y=\rho\sin{\phi}\sin{\theta}$

$z=\rho\cos{\phi}$

and

$\rho^2=x^2+y^2+z^2$

How to convert between spherical coordinates and rectangular coordinates

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Converting a rectangular coordinate point to a spherical coordinate point, then a spherical coordinate point to a rectangular coordinate point

Let’s try an example where we convert rectangular coordinates to spherical coordinates.

Example

Convert the rectangular coordinate point to a spherical coordinate point.

$(0,0,1)$

We’ll start by plugging $(0,0,1)$ into $\rho^2=x^2+y^2+z^2$.

$\rho^2=0^2+0^2+1^2$

$\rho^2=1$

Since $\rho\geq 0$, $\rho=1$

We’ll plug $(0,0,1)$ and $\rho=1$ into $z=\rho\cos{\phi}$ to solve for $\phi$.

$z=\rho\cos{\phi}$

$1=(1)\cos{\phi}$

$\cos{\phi}=1$

$\phi=0,\ 2\pi$

Since $0\leq\phi\leq\pi$, $\phi=0$

We’ll plug $(0,0,1)$, $\rho=1$, and $\phi=0$ into $y=\rho\sin{\phi}\sin{\theta}$ to solve for $\theta$.

$y=\rho\sin{\phi}\sin{\theta}$

$0=(1)\sin{(0)}\sin{\theta}$

$0=(1)(0)\sin{\theta}$

Since we have $0$ on the right-hand side, $\theta$ could be any value and the equation would still be true. This makes sense, since the given point is on the $z$-axis, and $\theta$ is the angle between $r$ (the shadow of the line connecting $(\rho,\theta,\phi)$ to the origin) and the positive direction of the $x$-axis.

Since it can be any value, let’s just choose $\theta=0$.

Putting these values together, we can say that the spherical coordinate $(1,0,0)$ is the same as the rectangular coordinate $(0,0,1)$.

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Let’s try an example where we convert spherical coordinates to rectangular coordinates.

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Like cartesian (or rectangular) coordinates and polar coordinates, spherical coordinates are just another way to describe points in three-dimensional space.

Example

Convert the spherical coordinate point to a rectangular coordinate point.

$\left(1,\pi,\frac{\pi}{2}\right)$

We know that

$\rho=1$

$\phi=\frac{\pi}{2}$

$\theta=\pi$

Plugging these into the conversion formulas, we get

$x=\rho\sin{\phi}\cos{\theta}$

$x=(1)\sin{\frac{\pi}{2}}\cos{\pi}$

$x=(1)(1)(-1)$

$x=-1$

and

$y=\rho\sin{\phi}\sin{\theta}$

$y=(1)\sin{\frac{\pi}{2}}\sin{\pi}$

$y=(1)(1)(0)$

$y=0$

and

$z=\rho\cos{\phi}$

$z=(1)\cos{\frac{\pi}{2}}$

$z=(1)(0)$

$z=0$

The rectangular coordinate $(-1,0,0)$ is the same as the spherical coordinate $(1,\pi,\pi/2)$.

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