the t in spherical t-design

Question

I would like to understand exactly what the "t" in a spherical t-design means.

I'm facing the following questions so I can understand the concept:

1. does "t" represent some kind of order? And if so,
2. what does that order mean? for example:
3. What's the difference between a t = 3 vs. t = 5 design?
4. what does the value of t signify/mean regarding the physical description of the points on a sphere?
5. Am I correct in assuming that for designs to distribute nodes on the surface of a sphere I'm interested in spherical t-designs on $S^2$
6. Are the polynomials that the math descriptions give (refer to link provided above) the spherical harmonics in my case?

For context, I'm designing a layout for an Ambisonic decoding system based on spherical geometry. It is discussed that using spherical t-designs for locating nodes on the sphere are best for maintaining energy across the sphere. I understand this. Where I get lost is that there of lots of different t-designs that have been developed, and trying to determine which would be best for the job.

Thank you for assisting me understand just what the math behind t-design means in a practical sense. :)

Answer 1

As taken from your link, here is the definition.

Definition: A set of $N$ points is called a spherical $t$-design if the integral of any polynomial of degree at most $t$ over the sphere is equal to the average value of the polynomial over the set of $N$ points.

Let me fill in some of the gaps in this definition. They are (mostly) interested in the unit sphere in 3D space $$\mathbb{S}^2=\{(x,y,z)\in\mathbb{R}^3\mid x^2+y^2+z^2=1\}\subset\mathbb{R}^3,$$ (although some other sources, if you search the literature, rescale the sphere so that it has surface area $1$). Now we consider all polynomials on $\mathbb{R}^3$ with degree at most $t$. For example, if $t=3$ these are all functions of the form $$P(x,y,z)=a+b_1x+b_2y+b_3z+c_1xy+c_2yz+c_3zx+d_1x^2+d_2y^2+d_3z^2$$$$+e_1x^2y+e_2y^2z+e_3z^2x+e_4xy^2+e_5yz^2+e_6zx^2+fxyz,$$ where $a, b_i,c_i,d_i,e_i,f\in\mathbb{R}$ are constants. Given such a polynomial, we can compute its average value over the surface of the sphere by integrating $$A(P)=\frac{1}{\textrm{area}(\mathbb{S}^2)}\int_{\mathbb{S}^2}P(x,y,z)\;\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z.$$ Computing $A$ directly might be difficult, so we might instead want to approximate $A$ by just computing it at finitely many points $X=\{(x_1,y_1,z_1),\dots,(x_N,y_N,z_N)\}\subset\mathbb{S}^2$: $$A(P)\simeq\frac{1}{N}\sum_{i=1}^NP(x_i,y_i,z_i)$$.

Now, the definition says that $X$ is a spherical $t$-design if for every polynomial with degree at most $t$ then this approximation exactly equals $A(P)$. So we have two variables: $N$ and $t$ and they work against each other. The larger $N$ is the more points there are and so the better we can make the approximation - conversely, the larger $t$ is the more complicated the polynomials $P$ can be, and so the harder they become to approximate. That is why in the list at the link the value of $N$ increase as $t$ increases.

The way this might be useful in practice is as follows. Suppose you have some collection $\mathfrak{F}$ of functions (not necessarily polynomials) on $\mathbb{S}^2$ and you want to efficiently approximate $A(f)$ for any $f\in \mathfrak{F}$. Suppose further that you do not mind approximating the functions in $\mathfrak{F}$ by polynomials of some bounded degree $t$. Then for any $f\in\mathfrak{F}$ there is some $P(x,y,z)$ such that $A(f)\simeq A(P)$ and you just have to compute $A(P)$. With a spherical $t$-design $X$ you can compute this efficiently and accurately.

So to answer your questions:

1. $t$ represents the maximum "complexity" of the functions (polynomials) whose average value over the sphere we want to compute.
2. Specifically, the degree is the maximum sum $k_x+k_y+k_z$ where $cx^{k_x}y^{k_y}z^{k_z}$ is a term in the polynomial for $c\in\mathbb{R}\backslash\{0\}$.
3. Every $t=5$ design is a $t=3$ design since if a design can be used to compute $A(P)$ for $P$ of degree at most $5$, then it can certainly be used for polynomials of degree at most $3$. The reverse is not true, a polynomial of degree $5$ may be too complicated for $A(P)$ to be computed with set of points only designed to deal with polynomials of degree $3$.
4. It's difficult, I think, to say something concrete, but in general the higher $t$ is, the more points there will be, and so the more evenly spaced out they will be. If you look at the library of 3D points at that link then for $t=1$ and $N=2$, $X$ is two opposite points, but for $t=2$ and $N=4$, $X$ is the set of vertices of an inscribed tetrahedron.
5. Yes, this is the setting I've focussed on, but everything generalises immediately to unit spheres in higher dimensions.
6. I'm by no means certain, but I suspect the answer is no.