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Parametric Families of Minimal Positive Cubature Formulas of Degrees 4 and 6 on the 2-Sphere

  • D. V. Gorbachev
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Abstract

We study minimal positive cubature formulas on the two-dimensional sphere of degrees four and six, with ten and eighteen nodes, respectively. We prove the regularity of Popov's symmetric formulas and establish that their local families of deformations are two-dimensional, up to rotations. We obtain algebraic relations between the weights and pairwise inner products of the nodes. For degree four, we construct a one-parameter family with half-turn symmetry connecting the formulas of Popov and Heo--Xu. We also give a self-contained proof that eighteen nodes are minimal for degree six.

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Cite

BibTeXRIS
BibTeX
@misc{gorbachev2026parametric,
  title = {{Parametric Families of Minimal Positive Cubature Formulas of Degrees 4 and 6 on the 2-Sphere}},
  author = {D. V. Gorbachev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14287},
  eprint = {2610.14287},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14287},
}
RIS
TY  - UNPB
TI  - Parametric Families of Minimal Positive Cubature Formulas of Degrees 4 and 6 on the 2-Sphere
AU  - D. V. Gorbachev
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.14287
UR  - https://arxiv.org/abs/2610.14287
AB  - We study minimal positive cubature formulas on the two-dimensional sphere of degrees four and six, with ten and eighteen nodes, respectively. We prove the regularity of Popov's symmetric formulas and establish that their local families of deformations are two-dimensional, up to rotations. We obtain algebraic relations between the weights and pairwise inner products of the nodes. For degree four, we construct a one-parameter family with half-turn symmetry connecting the formulas of Popov and Heo--Xu. We also give a self-contained proof that eighteen nodes are minimal for degree six.
ER  -