Семейством точек называется нумерованный набор вершин элементов сетки в пространстве R𝑛. Таковым набором может быть набор вершин симплекса сетки, набор вершин пары смежных симплексов сетки и др. Посредством введенной в ранних работах метрики в пространстве таких семейств задача вычисления качества элемента сетки (например, мера отличия симплекса от вырожденного) может быть сведена к задаче вычисления расстояния между некоторым семейством и некоторым множеством семейств. Предлагается унифицированная схема вычисления такого расстояния, которую можно рассматривать как обобщение схемы исследования на экстремум гладкой функции.

The paper discusses the problem of calculating the quality characteristic of elements of a triangulation grid (in multidimensional space). We call quality the degree of difference between a mesh element in a sense critical. For example, the difference between a triangle and a degenerate triangle. To calculate this value, the following construction is used. Mesh element vertices are treated as a numbered set of 𝑋 — points point family. In the space of such sets, the metric proposed in early works is introduced. As a result, the problem boils down to calculating the value of dist(𝑋,𝒴) — the distance between a set 𝑋 and a set 𝒴 defined by critical elements. In early works, the problem was solved through the empirical classification of families 𝑌 ∈ 𝒴 and excluding those that are not known to be closest to 𝑋. Namely, such that the offset some point in the 𝑌 family is given by the 𝑌 ′ ∈ 𝒴 family, which is closer to 𝑋. At the same time, empirical classification led to a large amount of calculations. This work gives a standardized classification of the families of the set 𝒴, allowing you to accurately describe the set of 𝒴− ⊂ 𝒴 of such families for which The specified conversion is not possible. That is, 𝒴− is approved dist(𝑋,𝒴) = dist(𝑋,𝒴−) . In addition, a diagram is given for constructing families of the set 𝒴− — analogues of the points of the assumed extremum in the extremum research problem of the function of numerical variables.