# What is the parametric equation of a sphere?

May 30, 2016

$\left(x , y , z\right) = \left(\rho \cos \theta \sin \phi , \rho \sin \theta \sin \phi , \rho \cos \phi\right)$

#### Explanation:

One common form of parametric equation of a sphere is:

$\left(x , y , z\right) = \left(\rho \cos \theta \sin \phi , \rho \sin \theta \sin \phi , \rho \cos \phi\right)$

where $\rho$ is the constant radius, $\theta \in \left[0 , 2 \pi\right)$ is the longitude and $\phi \in \left[0 , \pi\right]$ is the colatitude.

Since the surface of a sphere is two dimensional, parametric equations usually have two variables (in this case $\theta$ and $\phi$).

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Footnote

If you are determined to have a parametric equation with just one variable $t$ (say), then it is possible. For example, you can construct a surjection from the interval $\left[0 , 1\right]$ onto the rectangle $\left[0 , 2 \pi\right] \times \left[0 , \pi\right]$ and hence onto the surface of the sphere.

Such a surjection can even be made continuous. I'll see if I can put together a simple short formulation.