Q:

Derive parametric equations for sphere

How do you derive the parametric equations for a sphere?
\begin{align}
x & = r \cos(\theta)\sin(\varphi), \\
y & = r \sin(\theta)\sin(\varphi), \\
z & = r \cos(\varphi),
\end{align}
where $\theta$ is from $0$ to $2\pi$ and $\varphi$ is from $0$ to $\pi$.
There are no good explanations online.

A:

If you're familiar with surfaces of revolution, the derivation is easy. A circle that is rotated around a diameter generates a sphere.
The parametric equations for a surface of revolution are:
$$
\left(f(u)\cos v, f(u)\sin v, g(u)\right)
$$
Where $\left(f(u), g(u)\right)$ are the parametric equations of the rotated curve. For a circle, they are $\left(r \cos u, r \sin u\right)$. Therefore, the parametric equations of a sphere are:
$$
\left(r \cos u \cos v, r \cos u \sin v, r \sin u\right)
$$
Change the variables from $(u, v)$ to $(\frac{\pi}{2} - \varphi, \theta)$ to get the equations in your question.

