The goal of this post is to create a "good" algorithm for sampling and arranging points on the $n$-sphere. We find the $\epsilon$-covering number of the $n$-sphere and arrange the points in a Hamiltonian path of small pairwise consecutive distance. This post relates to several previous posts:
\[
\begin{array}{r c l}
f\ :\ [0,2\pi]^{n-1}\times[0,\pi) & \to & \R^{n+1}, \\
(\alpha_1,\dots,\alpha_n) & \mapsto & \big(\cos(\alpha_1), \\ && \sin(\alpha_1)\cos(\alpha_2),\\ && \vdots\\ && \sin(\alpha_1)\cdots\sin(\alpha_{n-1})\cos(\alpha_n), \\ && \sin(\alpha_1)\cdots\sin(\alpha_{n-1})\sin(\alpha_n)\big).
\end{array}
\]
Adapting the main Proposition from the "Sampling points" post, we have following proposition.
Proposition: The probability density function $g_n:[0,2\pi]^{n-1}\times[0,\pi] \to \R_{\geqslant0}$, defined as
\[
g_n(\alpha_1,\dots,\alpha_n)=\frac{\prod_{k=1}^{n-1}|\sin^{n-k}(\alpha_k)|}{2^{n-1}\pi\prod_{k=1}^{n-1}\int_0^\pi \sin^{n-k}(\alpha_k)\ d\alpha_k},
\]
is uniform on the natural embedding of the unit $n$-sphere $S^n$ in $\R^{n+1}$.
The denominator of $g_n$ does not seem to have closed form, though the ratios between consecutive terms are given by the denominators of $\Gamma(\frac{\ell+3}2)/\Gamma(\frac{\ell+2}2)$ and $\ell!!/(\ell+1)!!$, with appropriate powers of $\pi$. The first few terms of this sequence are
\[
4\pi,4\pi^2,\frac{32}3\pi^2,8\pi^3,\frac{256}{15}\pi^3,\frac{32}3\pi^4,\frac{2048}{105}\pi^4,\dots.
\]
Next, recall the $n$-surface of an $n$-sphere and $k$-volume of a $k$-ball are
\[
\text{surf}(n,r) = \frac{2\pi^{(n+1)/2}r^n}{\Gamma((n+1)/2)},\hspace{2cm}
\text{vol}(k,r) = \frac{\pi^{k/2}r^k}{\Gamma((k+2)/2)}.
\]
Adapting Proposition 3.2 of Niyogi, Smale and Weinberger, similarly to the "Reconstructing a manifold" post, we have the following proposition.
Proposition: A collection of $N$ points sampled uniformly from $S^n$ is $\epsilon$-dense in $S^n$ with certainty $1-\delta$, given
\[
N \geqslant \frac{\text{surf}(n,1)}{(1-\frac{\epsilon^2}{16})^{n/2}\text{vol}(n,\frac\epsilon2)}\log\left(\frac{\text{surf}(n,1)}{\delta(1-\frac{\epsilon^2}{64})^{n/2}\text{vol}(n,\frac\epsilon4)}\right).
\] Bauer and Polthier sample points "evenly" on the 2-hemisphere and then connect them with a winding path, which winds around the hemisphere 6 times. Generalizing this approach, suppose we wanted to have a path that wind around the $n$-sphere $\ell$ times and has a small distance between consecutive vertices of the path. The following algorithm describes one way of doing this.
Since the sample space is $[0,2\pi]^{n-1}\times[0,\pi]$, finding the appropriate points in the nested for loop is very easy. We conclude with an experimental example with $n=2$, $\ell=12$, $\epsilon=.1$, and $\delta=.01$. We must sample at least 87 points, and we do so below.
Example: To demonstrate the results of the SpherePathFinder algorithm, we sample 100, 300, and 600 points on the 2-sphere. Only the paths are shown, which wind around 12 times. The range of distances $d$ between consecutive ordered points is also given, with an average $\widetilde d$.
As $N$ increases and the winding number stays the same, the path gets more and more jagged. To make the path smoother, we would need to increase the number of times the path winds around the sphere.
References: Bauer and Polthier (Detection of Planar Regions in Volume Data for Topology Optimization), Niyogi, Smale, and Weinberger (Finding the homology of submanifolds with high confidence from random samples), Sloane (OEIS A036069, A004731), Wikipedia (article "N-sphere")
- 2017-02-12: Generalizing planar detection to $k$-plane detection
- 2016-12-22: Sampling points uniformly on parametrized manifolds
- 2016-05-26: Reconstructing a manifold from sample data, with noise
\[
\begin{array}{r c l}
f\ :\ [0,2\pi]^{n-1}\times[0,\pi) & \to & \R^{n+1}, \\
(\alpha_1,\dots,\alpha_n) & \mapsto & \big(\cos(\alpha_1), \\ && \sin(\alpha_1)\cos(\alpha_2),\\ && \vdots\\ && \sin(\alpha_1)\cdots\sin(\alpha_{n-1})\cos(\alpha_n), \\ && \sin(\alpha_1)\cdots\sin(\alpha_{n-1})\sin(\alpha_n)\big).
\end{array}
\]
Adapting the main Proposition from the "Sampling points" post, we have following proposition.
Proposition: The probability density function $g_n:[0,2\pi]^{n-1}\times[0,\pi] \to \R_{\geqslant0}$, defined as
\[
g_n(\alpha_1,\dots,\alpha_n)=\frac{\prod_{k=1}^{n-1}|\sin^{n-k}(\alpha_k)|}{2^{n-1}\pi\prod_{k=1}^{n-1}\int_0^\pi \sin^{n-k}(\alpha_k)\ d\alpha_k},
\]
is uniform on the natural embedding of the unit $n$-sphere $S^n$ in $\R^{n+1}$.
The denominator of $g_n$ does not seem to have closed form, though the ratios between consecutive terms are given by the denominators of $\Gamma(\frac{\ell+3}2)/\Gamma(\frac{\ell+2}2)$ and $\ell!!/(\ell+1)!!$, with appropriate powers of $\pi$. The first few terms of this sequence are
\[
4\pi,4\pi^2,\frac{32}3\pi^2,8\pi^3,\frac{256}{15}\pi^3,\frac{32}3\pi^4,\frac{2048}{105}\pi^4,\dots.
\]
Next, recall the $n$-surface of an $n$-sphere and $k$-volume of a $k$-ball are
\[
\text{surf}(n,r) = \frac{2\pi^{(n+1)/2}r^n}{\Gamma((n+1)/2)},\hspace{2cm}
\text{vol}(k,r) = \frac{\pi^{k/2}r^k}{\Gamma((k+2)/2)}.
\]
Adapting Proposition 3.2 of Niyogi, Smale and Weinberger, similarly to the "Reconstructing a manifold" post, we have the following proposition.
Proposition: A collection of $N$ points sampled uniformly from $S^n$ is $\epsilon$-dense in $S^n$ with certainty $1-\delta$, given
\[
N \geqslant \frac{\text{surf}(n,1)}{(1-\frac{\epsilon^2}{16})^{n/2}\text{vol}(n,\frac\epsilon2)}\log\left(\frac{\text{surf}(n,1)}{\delta(1-\frac{\epsilon^2}{64})^{n/2}\text{vol}(n,\frac\epsilon4)}\right).
\] Bauer and Polthier sample points "evenly" on the 2-hemisphere and then connect them with a winding path, which winds around the hemisphere 6 times. Generalizing this approach, suppose we wanted to have a path that wind around the $n$-sphere $\ell$ times and has a small distance between consecutive vertices of the path. The following algorithm describes one way of doing this.
Algorithm: SpherePathFinder
Input: Positive integers $n,\ell$ and real numbers $\epsilon,\delta\in (0,1)$
Output: A path on $S^n$ that winds around $\ell$ times, whose vertices are $\epsilon$-dense on $S^n$ with certainty $1-\delta$
Sample $\lceil N\rceil$ points on $[0,2\pi]^{n-1}\times[0,\pi]$ according to $g_n$ in a set $X$
Initiate an empty path $P=()$
for $k_n\in\{1,\dots,\ell\}$:
for $k_{n-1}\in\{1,\dots,2\ell\}$:
$\vdots$
for $k_2\in\{1,\dots,2\ell\}$:
Set $L=\{\alpha\in X\ :\ \alpha_n\in[(k_n-1)\frac\pi\ell,k_n\frac\pi\ell], \alpha_{n-t}\in[(k_{n-t}-1)\frac{2\pi}{2\ell},k_{n-t}\frac{2\pi}{2\ell}],1<t<n-1\}$
Order $L$ by increasing values of $\alpha_1$
Append $L$ to the end of $P$ and set $X=X\setminus L$
Return $P$
Since the sample space is $[0,2\pi]^{n-1}\times[0,\pi]$, finding the appropriate points in the nested for loop is very easy. We conclude with an experimental example with $n=2$, $\ell=12$, $\epsilon=.1$, and $\delta=.01$. We must sample at least 87 points, and we do so below.
Example: To demonstrate the results of the SpherePathFinder algorithm, we sample 100, 300, and 600 points on the 2-sphere. Only the paths are shown, which wind around 12 times. The range of distances $d$ between consecutive ordered points is also given, with an average $\widetilde d$.
As $N$ increases and the winding number stays the same, the path gets more and more jagged. To make the path smoother, we would need to increase the number of times the path winds around the sphere.
References: Bauer and Polthier (Detection of Planar Regions in Volume Data for Topology Optimization), Niyogi, Smale, and Weinberger (Finding the homology of submanifolds with high confidence from random samples), Sloane (OEIS A036069, A004731), Wikipedia (article "N-sphere")