What is the parametric equation of a sphere?

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What is the parametric equation of a sphere?

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Answer 1 · 2016-05-30T20:34:22

$(x, y, z) = (ρ cos θ sin ϕ, ρ sin θ sin ϕ, ρ cos ϕ)$ $(x, y, z) = (rho cos theta sin phi, rho sin theta sin phi, rho cos phi)$

Explanation:

One common form of parametric equation of a sphere is:

$(x, y, z) = (ρ cos θ sin ϕ, ρ sin θ sin ϕ, ρ cos ϕ)$ $(x, y, z) = (rho cos theta sin phi, rho sin theta sin phi, rho cos phi)$

where $ρ$ $rho$ is the constant radius, $θ \in [0, 2 π)$ $theta in [0, 2pi)$ is the longitude and $ϕ \in [0, π]$ $phi in [0, pi]$ is the colatitude.

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

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Footnote

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

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What is the parametric equation of a sphere?

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