# Unfolding the $n$-dimensional sphere [duplicate]

Is there an extension to $n$ dimensions of the usual spherical coordinates mapping a three-dimensional sphere to a two-dimensional rectangle?

If you know the sum of the squares of two real numbers, say $x_1^2+x_2^2=A^2$, then you can parametrize the solutions like $x_1=A\cos t,$ $x_2=A\sin t$, with parameter $t\in(-\pi,\pi]$ (or any other interval of length $2\pi$ of your choice).

When there are three variables, $x_1^2+x_2^2+x_3^2=A^2$, you can immediately deduce that $x_3=A\cos t_2$ for some value of $t_2\in[0,\pi]$. As a consequence of the ever useful identity $$\cos^2t+\sin^2t=1$$ you know that, given this value of $x_3$, you have $x_1^2+x_2^2=A'^2$ with $A'=A\sin t_2$.

When there are more than three coordinates with a known sum of squares, say $x_1^2+x_2^2+\cdots+x_n^2=A^2$, then you can first split them into two groups any which way you want in such a way that there are at least two coordinates in both groups. For example, we know that there exists a unique number $u\in[0,\pi/2]$ such that $$x_1^2+x_2^2=A^2\cos^2u,\qquad x_3^2+x_4^2+\cdots+x_n^2=A^2\sin^2u.$$ As you then "know" the sums of squares of the subsets of coordinates, you can proceed using one of the listed techniques.

This gives several recursive methods for introducing generalized spherical coordinates. If there are $n$ variables, there will be $n-1$ coordinates. Some (like $u$ above) with a range $[0,\pi/2]$, some (like $t_2$) have range $[0,\pi]$, and some will have range $[0,2\pi)$ (like $t_1$ above).

Which way you should go probably depends on other needs. You can also split the coorinates off one at the time (using a bunch of $t_2$ like parameters). This will have the consequence that the number of parameters a cartesian coordinate depends on will vary more. If you keep splitting the coordinates into two roughly equal size subsets, then it will be more balanced. Your choice!

• I would like to use this method to find the minimum/maximum/saddle points of $u^\top K u$ where $u$ satisfies $u^\top M u = \gamma$ and where $K$ and $M$ are two square matrices of dimension $n \times n$. – pluton Jun 29 '13 at 9:12

This can be tricky, and it depends on why you are unfolding the n-sphere. Here is how a 4d planet unfolds.

In four dimensions, a planet would rotate along clifford-parallels. So the clifford-lines are lines of longitude, and there is the lattitude that would correspond to the gimble that holds a globe. In four dimensions, the lattitude forms a little 3d sphere of diameter 1.

When the sun does not follow one of thes parallels, it is opposite-parallel to exactly one clifford-line. The actual points where the sun is directly overhead, is then a line of 'lattitude' on this sphere-let, and the general climate corresponds to this lattitude (ie S = equator = 0 degrees, N = polar = 90 degrees.

The lines of 'longitude' on this sphere-let is a season-zone. We have two season-zones, the north is 6 months behind the south, for example. There this becomes a full season-wheel, so there is somewhere at every season.

The unfolding then gives a cuboid, of these meanings.

$2\pi=360°$ makes the lines of longitude, the position of the sun in the day

$\pi=180°$ makes the lines of the year-cycle, the position is the season of the year

$1 = 90°$, is the climate, from equatorial to tropics.

There are other ways to unfold a sphere. One can always do what Buckminster Fuller did, and put it on a polytope, and unfold that.