Geodesic
Methods for constructing invariant cubature formulas\\ for integrals over the surface of a torus in ${\mathbb R}^3$
Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 146, pp. 218-228

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The article considers the question of constructing cubature formulas for the surface of a torus $T$ in ${\mathbb R}^3$, invariant under the group $G$ generated by reflections of $T$ into itself. For currently known invariant cubature formulas with a degree of accuracy greater than $3$, the number of nodes significantly exceeds the minimum possible. The article proposes invariant cubature formulas of degree $5$ and $7$ for the surface of a torus with a number of nodes close to the minimum. Tables of values of nodes and coefficients of the constructed cubature formulas are given. The dependence of these values on the ratio of the radii of the guide and generatrix of the torus circles is studied. For construction, the method of invariant cubature formulas is used, based on the theorem of S. L. Sobolev.
Keywords: cubature formulas, group of torus to self transformations
Mots-clés : torus, invariant polynomials 
@article{VTAMU_2024_29_146_a7,
     author = {I. M. Fedotova and M. I. Medvedeva and A. S. Katsunova},
     title = {Methods for constructing invariant cubature formulas\\ for integrals over the surface of a torus in ${\mathbb R}^3$},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {218--228},
     publisher = {mathdoc},
     volume = {29},
     number = {146},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2024_29_146_a7/}
}
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I. M. Fedotova; M. I. Medvedeva; A. S. Katsunova. Methods for constructing invariant cubature formulas\\ for integrals over the surface of a torus in ${\mathbb R}^3$. Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 146, pp. 218-228. http://geodesic.mathdoc.fr/item/VTAMU_2024_29_146_a7/