Rob Womersley

Last updated 05-Jun-2015

• Unit sphere S² in R³ has area |S²| = 4 π.
• The space P_t of polynomials of degree at most t on S² has dimension
  d_t = (t+1)².

• Efficient spherical t-designs are sets of N ~ t²/2 points x_j, j = 1,...,N on S² such that equal weight cubature with these nodes is exact for all polynomials in P_t.
• There are several equivalent characterizations of a spherical t-design: See here for more details.
  • The Weyl sums
    r_{l, k}(X_N)   :=   Σ_j=1,...,N Y_{l, k}(x_j) = 0
    for k = 1,...,2l+1, and l = 1, 2, 3, ..., t.
  • A potential function
    A_{t, N, ψ}(X_N) = (1/N²) Σ_i Σ_j ψ(x_i^T x_j)
    where ψ is a polynomial of degree t with strictly positive Legendre coefficients.
  • The potential reported below uses
    ψ(z) = z^t + z^t-1 - a₀
    where a₀ is the zeroth Legendre coefficient.
• These symmetric spherical designs have the number of points
  N = ceil((t+1)²/2)+1
  except for t = 3, 5, 7, 9, 11.

• The geodesic distance between two points x and y on the unit sphere S² is
  dist(x, y) = cos^-1(x^T y).
• The spherical cap with centre y and radius α is
  C(y, α) = {x on S²: dist(x, y) ≤ α }

• The minimum separation is
  min_{i ≠ j}   dist(x_i, x_j)
  is twice the packing radius for identical caps with centres x_j, j = 1,...,N.
• The mesh norm or covering radius is
  h = max_{x in S²}   min_j=1,...,N   dist(x, x_j).
  This is also the radius for covering the sphere by spherical caps with centers x_j for j = 1,...,N.
• The mesh ratio is
  ρ_N = covering radius / packing radius
    = 2 mesh norm / separation ≥ 1

• A sequence if point sets X_N is quasi-uniform if the mesh ratio if uniformly bounded.
• For a quasi-uniform sequence of point sets both the mesh norm and the covering radius have the optimal order N^-1/2
• There are many other measures related to the geometric quality of the point set:
  • Riesz s-energy, in particular the Coulomb (s=1) and log (s=0) cases.
  • Discrepancy, for example the spherical cap discrepancy or the Cui and Freeden discrepancy.
  • Volume of the convex hull of the points.
  • Properties of the Voronoi cells, such as area of largest and smallest Voronoi cells.

Caveat: All points are only numerical spherical designs within the limits of IEEE double precision. There are many approximate spherical designs with these numbers of points, but different geometric properties.

• For each point set, the text file has three items per row: the x_j, y_j, and z_j Cartesian coordinates in [-1, 1] for the point x_j = (x_j, y_j, z_j) on S².
• The equal cubature weights w_j = |S²|/N for j = 1,...,N are not included in the files.
• The number of rows is equal to the number of points N.
• All points are on the unit sphere so have |x_j|² = x_j² + y_j² + z_j² = 1 for j = 1,...,N.
• All criteria and rotationally invariant, so the point sets are normalised with the first point at the north pole and the second point on the prime meridian
• Symmetric (antipodal) spherical t-designs up to degree 297 are available here.
• The file names have three components: point set, degree, number of points.
• Points last updated 29-Nov-2012

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