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I am interested in evenly distributing points on the surface of spheres in dimensions 3 and higher.

This answer shows an excellent method called the Fibonacci lattice (also known as the Fibonacci spiral).

This excellent link from the comments shows us that in 3d cylindrical coordinates we can generate N points like this:

  • $P_i = \theta i$
  • $T_i = \sqrt{1-Z_i^2}$
  • $Z_i = (1-\frac{1}{N})(1-\frac{2i}{N-1}) $

There is an implementation of that algorithm in this answer that generates the following:

Sphere

Can this method be extended to dimensions higher than 3? Failing that, are there any other methods we know of for evenly distributing points on the surface of hyperspheres in N dimensions?

I am not interested in:

  • Creating a uniform random distribution on a hypersphere (like this), because I want the minimum distance between any two points to be as large as possible instead of being randomly distributed.
  • Particle repulsion simulation type methods, because they are hard to implement and take an extremely long time to run for large N.

I am aware that the distribution the Fibonacci lattice generates is not perfectly even, but other methods that generate a more even distribution are significantly more complicated and only offer a modest improvement.

My attempt so far:

I know that to extend the above equations to 4 dimensions I would have to add an additional term. This is where it gets confusing, but what I have worked out so far is:

  • Obviously the P term stays the same, it's just the angle.
  • T is just the radius from the centre, which I think would mostly stay the same (changed slightly just to take into account radius with the extra term).
  • Z looks a bit complicated, but really it's just giving an evenly distributed number between -1 and +1 for N samples.

The question then comes what to do with the additional term? Some possible options I've considered:

  • Add another Z type term where we just generate an even distribution between -1 and +1. So there are two terms generating even distributions between -1 and + 1, but it would be best to not have them "line up" which leads to the next option:
  • Treat Z and the new term as a two dimensional surface and try to generate an even distribution over this surface while keeping p the same and T still an expression for the radius.
  • Treat the new term as another golden angle type term so we have two golden angle type terms.

I think point 2 may be most likely.

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  • $\begingroup$ Perhaps reuse the "golden angle spiral trick" for 4-dimensional "cylindrical" coordinates? Like here it was done for a disc (2-d) and a sphere (3-d): blog.marmakoide.org/?p=1 . $\endgroup$
    – Vepir
    Jul 18 '19 at 8:38
  • $\begingroup$ Good link. That page explains the 2d and 3d case well. On that page in 3d, cylindrical coordinates they use p for the angle and z for the height, which as I understand is all you need to describe a cylinder in 3d (assuming the radius is 1). My confusion comes when I try to think about 4d cylindrical coordinates. I know I have to add another term along with the p and z, but do I just treat the additional term as another "height" coordinate? Does the expression for z stay the same? Having trouble working out the expression for the additional term. $\endgroup$
    – Karl
    Jul 18 '19 at 9:33
  • $\begingroup$ I've edited the question quite a bit, copying the equations from that link and my thoughts so far on extending those equations to 4 dimensions. $\endgroup$
    – Karl
    Jul 18 '19 at 10:43
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So, as you pointed out, the idea with Fibonacci lattice is based on the fact that when distributed uniformly, both the azimuthal angle $\phi$ and latitude sine (affine with $\cos\theta$) are uniform.

To distribute my points on the $d$ dimensional surface of the sphere $\mathbb{S}^d$ embedded in $d+1$ dimensions, first I need to introduce coordinates. Let's use the generalized polar coordinate system. $$(r, \theta^1, \theta^2, \cdots, \theta^d)$$ This is converted to Cartesian coordinates as $$x^{d+1}=r\cos\theta^d$$ $$x^{d}=r\sin\theta^d\cos\theta^{d-1}$$ $$\vdots$$ $$x^{1}=r\sin\theta^d\cdots\sin\theta^1$$ Now to produce points on $\mathbb{S}^d$, we need $r=1$. It remains to come up with some distribution over $\theta$ coordinates that induces a uniform surface measure on the sphere surface. $$\rho_{\vec\theta}(\vec\theta)=\rho_{\theta^d}(\theta^d)\rho_{\theta^{d-1}|\theta^d}(\theta^{d-1})\cdots\rho_{\theta^1|\theta^2\cdots\theta^d}(\theta^1)$$ An interesting feature of our polar coordinates is that fixing the last $k$ angular coordinates, induces the same coordinate system for a $d+1-k$ dimensional space, assuming one uses the redefinition $$r\rightarrow r\sin\theta^d\cdots\sin\theta^{d-k+1}$$ (Check!) But this means that the angles are independently distributed since the above redefinition only rescales the space and therefore leaves the angular distributions unchanged. We find $$\rho_{\vec\theta}(\vec\theta)=\prod_{\alpha=1}^d\rho_\alpha(\theta^\alpha)$$

It is not hard to prove that $$\rho_\alpha=\frac{1}{\sqrt\pi}\frac{\Gamma\big(\frac{\alpha+1}{2}\big)}{\Gamma\big(\frac{\alpha}{2}\big)}\sin^{\alpha-1}$$ Although for $\alpha=1$ this formula generates a uniform angle in $(0,\pi)$. We assume that this issue has been resolved manually.

Denote the C.D.F's with $$Y^\alpha(\theta^\alpha)=\int_0^{\theta^\alpha}\rho_\alpha(u)du$$ They satisfy (as they should) $$Y^\alpha(0)=0,\hspace{5mm}Y^\alpha(\pi)=1$$ So far we have proved that a point $\vec X$ will have uniform distribution on $\mathbb{S}^d$ if $\vec{X}=\vec{X}(1, \vec\theta)$ and $\vec\theta=\vec\theta(\vec Y)$ if and only if $$\vec Y\sim \textrm{Uni}\big([0,1]^d\big)$$

Remark: The map from $Y$ hypercube to the sphere is $\pi-$Lipschitz!

Now this was what you didn't want :)) A random point ensemble on the sphere. To make it Fibonacci-like, we won't be taking random independent points from the hypercube of $Y$'s. Instead for $n\in\big[N\big]$ generate the points $\vec{Y}_n$ with $$Y^d_n=\frac{n}{N+1}$$ $$Y^{d-1}_n=\{na_1\}$$ $$Y^{d-2}_n=\{na_2\}$$ $$\vdots$$ $$Y^1=\{na_{d-1}\}$$ For fixed irrational numbers $a_i$ satisfying $$\frac{a_i}{a_j}\not\in\mathbb{Q}\hspace{4mm}\forall i\neq j$$ A series of numbers that you can always use for $a$'s is $$\sqrt2, \sqrt3, \sqrt5, \sqrt7, \sqrt{11}, \sqrt {13}, \cdots$$

Example: $d=3$ with above $a$'s. $$t=\cos\psi$$ $$z=\sin\psi\cos\theta$$ $$y=\sin\psi\sin\theta\cos\phi$$ $$x=\sin\psi\sin\theta\sin\phi$$

The angles are given by $$\psi_n=f^{-1}\big(\frac{n\pi}{N+1}\big), \hspace{3mm} \hspace{1mm} f(x)\equiv x-\frac{1}{2}\sin2x$$ $$\theta_n=\arccos\big(1-2\{n\sqrt2\}\big)$$ $$\phi_n=2\pi\{n\sqrt3\}$$

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  • $\begingroup$ I have implicitly assumed that the vector sequence $\{n\vec u\}$ densely covers the unit hypercube if the components of the vector $\vec u$ are irrational and also mutually irrational. $\endgroup$
    – K. Sadri
    Jul 19 '19 at 14:59
  • $\begingroup$ Great, this looks like everything I need but some of the maths here goes above my level of understanding at the moment. You have the bounty unless anything better comes along. I plan to turn this into pseudo code that works in N dimensions as I'd really like to investigate it's properties when run for different parameters. If I get stuck I may post another question. Alternatively, if you're able to include any extra detail that may help with turning it into pseudo code then that would be greatly appreciated, although I understand that may involve a significant level of effort. $\endgroup$
    – Karl
    Jul 19 '19 at 15:29
  • $\begingroup$ So in the last section ψ, ϕ and θ are effectively Y3, Y2 and Y1? $\endgroup$
    – Karl
    Jul 19 '19 at 15:44
  • $\begingroup$ Nope, they are the angular coordinates. The Ys are the decimal parts $\endgroup$
    – K. Sadri
    Jul 19 '19 at 15:45
  • $\begingroup$ I'm too lazy to simulate. Will you please first do the d=2 case (The sphere) this time with $\sqrt2$ instead of the golden number? I will try to find good ways for numerically calculating the inverse functions that appear. Thank you for the nice question :)) $\endgroup$
    – K. Sadri
    Jul 19 '19 at 15:48
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First of all a note about cartesian, polar and cylindrical coordinates, in Euclidean space

  • in 1D, you have only one coordinate, so no distinction;
  • in 2D, you have either Cartesian $(x_1 , x_2)$ or polar $(r ; \alpha)$;
  • in 3D, you have Cartesian $(x_1 , x_2 , x_3)$, polar $(r ; \alpha_1,\alpha_2)$ or cylindrical in which two of the cartesian coordinates are replaced with a couple of 2D polar coordinates;
  • in 4D, you have the four cartesian, the "full-polar" (1 radius, 3 angles), and different cylindrical in between, for example 1D c+ 3D spherical, or two 2D polar or 2D c. + 2P polar.

In general it is clear that you can partition a n-D space into subspaces and use for each sub-space any coordinate system.

Let's concentrate on the 4D case. There the equation equation of a unit sphere is $$ x_{\,1} ^{\,2} + x_{\,2} ^{\,2} + x_{\,3} ^{\,2} + x_{\,4} ^{\,2} = 1\quad \left| {\; - 1 \le x_{\,k} \le 1} \right. $$

A section at constant $x_4$ is a sphere in the 3D space.

The surface area of a $n-1$-d sphere (embedded into a $n$D space) is $$ S_{\,n - 1} (r) = {{2\pi ^{\,\,n/2} } \over {\Gamma \left( {n/2} \right)}}r^{\,\,n - 1} \quad \Rightarrow \quad \left\{ \matrix{ S_{\,3} = S_{\,4 - 1} (1) = 2\pi ^{\,\,2} \hfill \cr S_{\,2} (r) = 4\pi r^{\,\,2} \hfill \cr} \right. $$ and we get therefrom the values for the spheres mentioned above.

We can think of various repartitions of the points between two different dimensions , as you did and as indicated in
the excellent answer by @ksadri.

But the spiral principle, as far as I understood from the link you cited, is to construct a 1D line of length $L$ which traverses our 3D surface uniformly, i.e. such that the ratio $dl/dS_3 = L/S_3$ among the length of the line and the area "spanned" by it is constant.

After that we drop the given $N$ points randomly uniformly over $L$.
This can be done in various ways, but the one suggested in the link is to use the equidistribution of the multiples of an irrational number, that is of the sequence $$ \left\{ {k\,\alpha } \right\} = \left\{ {\left\lfloor {{l \over L}N} \right\rfloor \,\alpha } \right\}\quad \left| {\;1 \le k \le N} \right. $$ where $\alpha$ is a chosen irrational number, and the curly brackets indicate the fractional part.

So let's try and create such a spiral.
We take the spherical coordinates $$ \left\{ \matrix{ x_{\,1} = \sin \varphi _{\,3} \sin \varphi _{\,2} \sin \varphi _{\,1} \hfill \cr x_{\,2} = \sin \varphi _{\,3} \sin \varphi _{\,2} \cos \varphi _{\,1} \hfill \cr x_{\,3} = \sin \varphi _{\,3} \cos \varphi _{\,2} \hfill \cr x_{\,4} = \cos \varphi _{\,3} \hfill \cr} \right.\quad \left| \matrix{ \;0 \le \varphi _{\,3} ,\varphi _{\,2} \le \pi \hfill \cr \;0 \le \varphi _{\,1} < 2\pi \hfill \cr} \right. $$ individuated by the three angles $ \varphi _{\,1}, \varphi _{\,2}, \varphi _{\,3}$.

We have better to parametrize the line in the parameter $t$ and figure it as a time.
With reference to the sketch

Sfera_nD_points_1

we assign to $\varphi _{\,3}$ a constant angular speed $\omega_3 = \pi /T$ $$ \varphi _{\,3} ' = \omega_3 = {\pi \over T}\quad \varphi _{\,3} = \pi {t \over T}\quad \left| {\;0 \le t \le T} \right. $$ The area of the spherical zone spanned by $\varphi_3$ in $dt$ will be $$ S_{\,2} (\sin \varphi _{\,3} ) = 4\pi \sin ^2 \varphi _{\,3} \,d\varphi _{\,3} = 4\pi ^2 \sin ^2 \varphi _{\,3} {{dt} \over T} $$ therefore if we assign to the component of speed in $S_2$ a value proportional to that, we will satisfy the requirement of "uniform coverage" (the spiral will undergo a number of "turns" proportional to the area). $$ w_{\,2} = c_2 \sin ^2 \varphi _{\,3} $$ After that we want again to repart $w_2$ in the same way between $$ v_{\,2} = \sin \varphi _{\,3} \,\omega _{\,2} \quad v_{\,1} = \sin \varphi _{\,3} \,\sin \varphi _{\,2} \,\omega _{\,1} $$ which means $$ \eqalign{ & \left\{ \matrix{ w_{\,2} ^2 = v_{\,2} ^2 + v_{\,1} ^2 \hfill \cr {{v_{\,1} } \over {v_{\,2} }} \propto {{2\pi \sin \varphi _{\,2} \;d\varphi _{\,2} } \over {d\varphi _{\,2} }} \hfill \cr} \right.\quad \Rightarrow \cr & \Rightarrow \quad \left\{ \matrix{ {{v_{\,1} } \over {v_{\,2} }} = {{\sin \varphi _{\,2} \,\omega _{\,1} } \over {\,\omega _{\,2} }} = c_{\,1} \sin \varphi _{\,2} \hfill \cr w_{\,2} ^2 = v_{\,2} ^2 + v_{\,1} ^2 \hfill \cr} \right.\quad \Rightarrow \cr & \Rightarrow \quad \left\{ \matrix{ {{v_{\,1} } \over {v_{\,2} }} = c_{\,1} \sin \varphi _{\,2} \hfill \cr c_{\,2} ^{\,2} \sin ^2 \varphi _{\,3} = v_{\,2} ^2 \left( {1 + c_{\,1} ^2 \sin ^2 \varphi _{\,2} } \right) \hfill \cr} \right.\quad \Rightarrow \cr & \Rightarrow \quad \left\{ \matrix{ \omega _{\,1} = c_{\,1} \,\,\omega _{\,2} \hfill \cr \omega _{\,2} = {{c_{\,2} \sin \varphi _{\,3} } \over {\sqrt {1 + c_{\,1} ^2 \,\sin ^2 \varphi _{\,2} } }}\quad \, \Rightarrow \quad \varphi _{\,2} ' = {{c_{\,2} \sin \left( {\pi {t \over T}} \right)} \over {\sqrt {1 + c_{\,1} ^2 \,\sin ^2 \varphi _{\,2} } }} \hfill \cr} \right. \cr} $$

The ode $$ y(t)'\sqrt {1 + c_{\,1} ^2 \,\sin ^2 y(t)} = g(t) $$ can be solved by noting that $$ y'{d \over {dy}}\int_{u = 0}^y {\sqrt {1\, + c_{\,1} ^2 \,\sin ^2 u} \;du} = {d \over {dt}}\int_{u = 0}^y {\sqrt {1\, + c_{\,1} ^2 \,\sin ^2 u} \;du} $$ thus $$ {d \over {dt}}\left( {{{\varphi _{\,2} \sqrt {1\, + c_{\,1} ^2 \,\varphi _{\,2} ^2 } } \over 2} + {{\ln \left( {\sqrt {1\, + c_{\,1} ^2 \,\varphi _{\,2} ^2 } + c_{\,1} \,\varphi _{\,2} } \right)} \over {2c_{\,1} }}} \right) = c_{\,2} \sin \left( {\pi {t \over T}} \right) $$ after which it is a matter of calculus, or numerical integration, to find $l(t)$.

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  • $\begingroup$ Thanks, another great answer. I don't fully understand all of the maths used here as I am more of a programmer than a mathematician. I'm doing some programming to investigate sphere packing which led me to post this question. Are you interested in helping me turn either of these two answers into pseudo code on stack overflow here?: stackoverflow.com/q/57123194 $\endgroup$
    – Karl
    Jul 20 '19 at 13:46

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