3
$\begingroup$

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.

$\endgroup$
1
  • 2
    $\begingroup$ In 3D, there are 2 systems that could be thought of as generalizations of polar coordinates, namely, cylindrical coordinates, and spherical coordinates. I imagine both generalize easily to $n$ dimensions. $\endgroup$ Commented Oct 8, 2016 at 4:42

1 Answer 1

Reset to default
4
$\begingroup$

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.

$\endgroup$
8
  • $\begingroup$ I suppose $\theta$ is the angle between $x,y$. What is $\phi$? The angle between $y,z$ or $x,z$? $\endgroup$ Commented Oct 8, 2016 at 4:53
  • $\begingroup$ Change that angle and see how the point moves. $\endgroup$ Commented Oct 8, 2016 at 4:54
  • $\begingroup$ $\phi$ is the angle between the $z$-axis and the $xy$-plane. $\endgroup$ Commented Oct 8, 2016 at 4:57
  • 2
    $\begingroup$ @JackyChong You make it sound like there's is a unique (or at least prevalent) choice of angles in each dimension $n$, which isn't necessarily the case. You may want to elaborate that point some more. $\endgroup$ Commented Oct 8, 2016 at 4:57
  • $\begingroup$ @dxiv I don't quite understand your question. $\endgroup$ Commented Oct 8, 2016 at 4:59

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.