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Hybrid spherical designs

Martin Ehler

TL;DR

The paper tackles the construction of explicit spherical $t$-designs beyond discrete point sets by introducing design curves and their hybrids with point sets. It develops a unified framework based on group orbits, invariant polynomials, and dual polytopes to achieve exact integration for higher degrees $s$ by combining geodesic cycles with vertex orbits, including a remarkable $t=19$ hybrid on $\mathbb{S}^3$. Core methods employ invariant theory (Molien series) to identify $G$-invariant polynomials and balance contributions from curves and points via a parameter $\beta$, enabling explicit, analytically provable designs. The results generate new families of $t$-design curves and strong hybrid designs in dimensions up to $3$ and extend to higher dimensions, with a general non-transitive framework to widen applicability for spherical cubature and mobile sampling contexts.

Abstract

Spherical $t$-designs are finite point sets on the unit sphere that enable exact integration of polynomials of degree at most $t$ via equal-weight quadrature. This concept has recently been extended to spherical $t$-design curves by the use of normalized path integrals. However, explicit examples of such curves are rare. We construct new spherical $t$-design curves for small $t$ based on the edges of a distinct subclass of convex polytopes. We then introduce hybrid $t$-designs that combine points and curves for exact polynomial integration of higher degree. Our constructions are based on the vertices and edges of dual pairs of convex polytopes and polynomial invariants of their symmetry group. A notable result is a hybrid $t$-design for $t=19$.

Hybrid spherical designs

TL;DR

The paper tackles the construction of explicit spherical -designs beyond discrete point sets by introducing design curves and their hybrids with point sets. It develops a unified framework based on group orbits, invariant polynomials, and dual polytopes to achieve exact integration for higher degrees by combining geodesic cycles with vertex orbits, including a remarkable hybrid on . Core methods employ invariant theory (Molien series) to identify -invariant polynomials and balance contributions from curves and points via a parameter , enabling explicit, analytically provable designs. The results generate new families of -design curves and strong hybrid designs in dimensions up to and extend to higher dimensions, with a general non-transitive framework to widen applicability for spherical cubature and mobile sampling contexts.

Abstract

Spherical -designs are finite point sets on the unit sphere that enable exact integration of polynomials of degree at most via equal-weight quadrature. This concept has recently been extended to spherical -design curves by the use of normalized path integrals. However, explicit examples of such curves are rare. We construct new spherical -design curves for small based on the edges of a distinct subclass of convex polytopes. We then introduce hybrid -designs that combine points and curves for exact polynomial integration of higher degree. Our constructions are based on the vertices and edges of dual pairs of convex polytopes and polynomial invariants of their symmetry group. A notable result is a hybrid -design for .

Paper Structure

This paper contains 18 sections, 12 theorems, 79 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Curves, designs, and group orbits
  3. Spherical design curves
  4. Orbits of finite subgroups of the orthogonal group
  5. Spherical $t$-design cycles from convex polytopes
  6. Euler cycles of convex polytopes induce spherical $t$-design cycles
  7. Edge-transitive convex polytopes in $\mathbb{R}^3$
  8. Further edge-transitive polytopes
  9. Hybrid spherical designs
  10. Introduction to hybrid designs
  11. Hybrid designs from pairs of dual polytopes
  12. Proof of Theorem \ref{['thm:main']}
  13. Combining orbits of points with orbits of geodesic arcs
  14. Hybrid designs and invariant polynomials on the sphere
  15. Hybrid designs in higher dimensions
  16. ...and 3 more sections

Key Result

Lemma 2.2

If $G\subseteq\mathcal{O}(d+1)$ is $t$-homogeneous and $\gamma_0\subseteq\mathbb{S}^d$ is a geodesic arc, then the orbit $(g \gamma_0)_{g\in G}$ induced by $\gamma_0$ satisfies the exact integration condition

Figures (3)

  • Figure 7: The factor $\beta(a)$ in \ref{['eq:rho']} balances the contribution of the south pole with the triangle at height $\cos(a)$ in the proof of Part (ii) of Proposition \ref{['prop:1-2-3']}. We obtain a hybrid $2$-design at $\hat{a}\approx 1.359$ with $\beta(\hat{a})\approx 0.249$.
  • Figure 8: The invariant polynomials $p_{3,A_3}\in\mathcal{H}^{A_3}_3$, $p_{4,B_3}\in\mathcal{H}^{B_3}_4$, and $p_{6,H_3}\in\mathcal{H}^{H_3}_6$ are shown with the rainbow color scheme that ranges from $\min p_{\mathop{\mathrm{inv}}\nolimits}$ (violet) to $\max p_{\mathop{\mathrm{inv}}\nolimits}$ (red). The vertices of the associated pairs of convex regular polytopes lie precisely at the minima and maxima, respectively. The polynomials $p_{3,A_3}$, $p_{4,B_3}$, $p_{6,H_3}$ are constant on the (green) spherical edges of the octahedron, cuboctahedron, icosidodecahedron, respectively.
  • Figure 9: The normalized Hausdorff measures of $V(\mathcal{H}^{A_3}_3)$, $V(\mathcal{H}^{B_3}_4)$, and $V(\mathcal{H}^{H_3}_6)$ provide exact integration on $\mathop{\mathrm{Pol}}\nolimits_{\leq 3}$, $\mathop{\mathrm{Pol}}\nolimits_{\leq 5}$, and $\mathop{\mathrm{Pol}}\nolimits_{\leq 9}$, respectively.

Theorems & Definitions (29)

  • Example 2.1: Figures \ref{['fig:octa standard']}, \ref{['fig:cubo intro']}, \ref{['ggg0']}: arrangement of great circles in $\mathbb{S}^2$
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Definition 4.1: hybrid $t$-designs
  • Lemma 4.2
  • Proposition 4.3
  • proof
  • Theorem 4.4
  • ...and 19 more