Analogue of spherical coordinates in $n$-dimensions

What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by

$$x_1=r \cos \theta$$ $$x_2=r \sin \theta$$

For $n=3$, the analogue would be the ordinary spherical coordinates $r,\theta ,\varphi$, related to the Cartesian coordinates $x_1,x_2,x_3$ by

$$x_1=r \sin \theta \cos \varphi$$ $$x_2=r \sin \theta \sin \varphi$$ $$x_3=r \cos \theta$$

So these are my questions: Is there an analogue, or several, to spherical coordinates in $n$-dimensions for $n>3$? If there are such analogues, what are they and how are they related to the Cartesian coordinates? Thanks.

• Hyperspherical coordinates are on Wikipedia.
– anon
Aug 9 '11 at 19:22
• See Knuth, Art of Computer Prog., Vol.2 3.3.1, Exercise 15. Jul 25 '18 at 14:28

These are hyperspherical coordinates. You can see an example of them being put to use in this answer.

• That's really funny, because I happen to read that answer just yesterday. And I was going to reference it for this question too. Aug 9 '11 at 19:29
• Right now, your answer looks like a "link only" (or citation only) answer. As the goal of MSE is to provide a more-or-less self-contained repository of questions and answers, it would be preferable if you expended some words to explain what is contained in those references and how it applies to the question being asked. Moreover, the link relates to a hypersphere, not hyperspherical coordinates, which are the scope of the question. Sep 23 '18 at 20:46
• @Anakhand: Thanks for your comment. As to the first link, I updated it -- the section it pointed to was renamed in the meantime. As to the second link: Did you notice that this leads to an answer on this site? I agree that an answer shouldn't consist entirely of external links; but this is an internal link (which I trust will be automatically updated, should the URL structure of the site ever be changed) which illustrates the definition and usage of hyperspherical coordinates and is part of the self-contained repository of questions and answers that we're building here. Sep 23 '18 at 20:50
• @joriki Thank you for updating the first link. As for the second link, I did in fact notice it leads to another answer on this site, but I still believe a word or two on this answer wouldn't hurt. In any case, I want to apologize for the comment, as it might have seemed rude---or it might have seen as wanting to "lecture" a more experienced user---, but it wasn't my intention. Sep 24 '18 at 8:27

Just look at n-sphere. A lecture note from Stony Brook is also available.

You can find it in Fock's paper (Fock, V. (1935).)or in some recent papers- like (Howard, S. "Fundamental Solution of Laplaces Equation in Hyperspherical Geometry.") or (Jing-Jing, F., Ling, H., & Shi-Jie, Y. (2011). Solutions of laplace equation in n-dimensional spaces. Communications in Theoretical Physics, 56(4), 623.) for advanced studies.

• Right now, your answer looks like a "link only" (or citation only) answer. As the goal of MSE is to provide a more-or-less self-contained repository of questions and answers, it would be preferable if you expended some words to explain what is contained in those references and how it applies to the question being asked. Apr 25 '18 at 17:38

I was trying to answer exercise 9 of $$I.5.$$ from Einstein gravity in a nutshell by A. Zee that I saw this question so what I am going to say is from this question. It is said that the d-dimensional unit sphere $$S^d$$ is embedded into $$E^{d+1}$$ by usual Pythagorean relation$$(X^1)^2+(X^2)^2+.....+(X^{d+1})^2=1$$. Thus $$S^1$$ is the circle and $$S^2$$ the sphere. A. Zee says we can generalize what we know about polar and spherical coordinates to higher dimensions by defining

$$X^1=\cos\theta_1\quad X^2=\sin\theta_1 \cos\theta_2,\ldots$$

$$X^d=\sin\theta_1 \ldots \sin\theta_{d-1} \cos\theta_d,$$

$$X^{d+1}=\sin\theta_1 \ldots \sin\theta_{d-1} \sin\theta_d$$

where $$0\leq\theta_{i}\lt \pi \,$$ for $$1\leq i\lt d$$ but $$0\leq \theta_d \lt 2\pi$$.

So for $$S^1$$ we just have ($$\theta_1$$):

$$X^1=\cos\theta_1,\quad X^2=\sin\theta_1$$

$$S^1$$ is embedded into $$E^2$$ and for the metric on $$S^1$$ we have: $$ds_1^2=\sum_1^2(dX^i)^2=d\theta_1^2$$
for $$S^2$$ we have ($$\theta_1, \theta_2$$) so for Cartesian coordinates we have:

$$X^1=\cos\theta_1,\quad X^2=\sin\theta_1\cos\theta_2,$$ $$\quad X^3=\sin\theta_1\sin\theta_2$$

and for its metric: $$ds_2^2=\sum_1^3(dX^i)^2=d\theta_1^2+\sin^2\theta_1 d\theta_2^2$$
for $$S^3$$ which is embedded into $$E^4$$ we have($$\theta_1,\theta_2,\theta_3$$):

$$X^1=\cos\theta_1,\quad X^2=\sin\theta_1\cos\theta_2,$$ $$\quad X^3=\sin\theta_1\sin\theta_2\cos\theta_3$$ $$\quad X^4=\sin\theta_1\sin\theta_2\sin\theta_3$$
$$ds_3^2=\sum_{i=1}^4(dX^1)^i=d\theta_1^2+\sin^2\theta_1 d\theta_2^2+sin^2\theta_1\sin^2\theta_2\,d\theta_3^2$$
Finally, it is not difficult to show the metric on $$S^d$$ will be:

$$ds_d^2=\sum_{i=1}^{d+1}(dX^1)^i=d\theta_1^2+\sin^2\theta_1 d\theta_2^2+sin^2\theta_1\sin^2\theta_2\,d\theta_3^2+\cdots+sin^2\theta_1\cdots\sin^2\theta_{d-1}\,d\theta_d^2$$