What is the parametric equation of a sphere?

1 Answer
May 30, 2016

Answer:

#(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.