Question

What is the parametric equation of a sphere?

Answer 1

`(x, y, z) = (rho cos theta sin phi, rho sin theta sin phi, rho cos phi)`

Explanation:

One common form of parametric equation of a sphere is:

`(x, y, z) = (rho cos theta sin phi, rho sin theta sin phi, rho cos phi)`

where `rho` is the constant radius, `theta in [0, 2pi)` is the longitude and `phi in [0, pi]` 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 `[0, 1]` onto the rectangle `[0, 2pi] xx [0, pi]` 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.