On the generalization of polar coordinates for $n-$dimensions?

Question

I am acquainted with polar coordinates and their reason of being, that is: Taking a vector in $2-$dimensions, we can normalize it:

$$\cfrac{1}{|(x,y)|}(x,y)=(x',y')$$

And we can clearly see that:

$$(x,y)=|(x,y)|(x',y')$$

And we can choose suitable $x',y'$ such as $x'=\cos \theta, y'=\sin \theta$ because $\cos^2\theta + \sin^2\theta=1$ and $|(x',y')|=1$. But how should we proceed for $(n>2)-$dimensions? For example, for $3-$dimensions, my guess is that it's a repetition of the previous argument, but now we must have $x'^2+y'^2+z'^2=1$.

My problem is:

• What will be the suitable functions for $x',y,'z'$ in terms of angles?

• What will be the suitable functions for $n-$dimensional polar coordinates?

• Can we find suitable functions such for some other parameter instead of angles?

I'm supposing that such generalization is possible, but I'm actually not sure it is. From the sources I've found, there is only for $2,3$ dimensions.

Answer 1

Let us consider case by case.

Case: $n=2$.

We have \begin{align} x=&\ r\cos\theta, \\ y=&\ r\sin\theta \end{align}

Case: $n=3$.

We have \begin{align} x=&\ r\cos\theta \sin\phi,\\ y=&\ r\sin\theta \sin\phi, \\ z=&\ r\cos\phi. \end{align}

Case: $n=4$.

We have

\begin{align} x_1=&\ r\cos\theta \sin\phi\ \sin \tau,\\ x_2=&\ r\sin\theta \sin\phi\ \sin\tau,\\ x_3=&\ r\cos\phi \sin\tau, \\ x_4=&\ r\cos\tau. \end{align}

Case: $n=5$.

We have \begin{align} x_1=&\ r\cos\theta \sin\phi\ \sin \tau \sin \psi,\\ x_2=&\ r\sin\theta \sin\phi\ \sin \tau \sin \psi,\\ x_3=&\ r\cos\phi \sin\tau \sin\psi, \\ x_4=&\ r\cos\tau \sin\psi.\\ x_5=&\ r\cos\psi \end{align}

... I hope you see the pattern.

Edit: Just to be clear. For each of the above cases, I have provided a parametrization of the $n$-spheres which respect the underlying euclidean metric.