I'm reading an article on convex geometry, and there is a part that attempts to calculate the volume of the Euclidean ball in $\mathbb{R}^n$. The first part of the argument talks about spherical polar coordinates in $\mathbb{R}^n$. I'm unable to understand a few things, which I have written down after quoting an excerpt:
We shall need to know the volume of the ball: call it $v_n$ . We can calculate the surface “area” of $B^n_2$ very easily in terms of $v_n$: the argument goes back to the ancients. We think of the ball as being built of thin cones of height $1$: see Figure 4, left. Since the volume of each of these cones is $1/n$ times its base area, the surface of the ball has area $nv_n$. The sphere of radius $1$, which is the surface of the ball, we shall denote $S_{n−1}$. To calculate $v_n$, we use integration in spherical polar coordinates. To specify a point $x$ we use two coordinates: $r$, its distance from $0$, and $θ$, a point on the sphere, which specifies the direction of $x$. The point $θ$ plays the role of $n − 1$ real coordinates. Clearly, in this representation, $x = rθ$: see Figure 4, right.
Questions:
- How does $\theta$ represent $n-1$ real coordinates?
- Why do we have $x = r\theta$ here? It's not as obvious as it is in $\mathbb{R}^3$.
Well.. a point on the $n$-dimensional unit sphere is a vector with $n$ real coordinates subjected to the constraint $\|\Theta\|=1$, which leaves only $n-1$ free parameters. For example, in 2D you have $x=r(\cos\theta,\sin\theta)$ (polar coordinates) so $\Theta=(\cos\theta,\sin\theta)$. In 3D you have $x=r(\cos\phi\cos\theta,\cos\phi\sin\theta,\sin\phi)$ (here $\theta$ and $\phi$ are spherical coordinates), etc. There are general "polar" coordinates in any dimension. You can check
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