DOI: 10.14489/vkit.2018.08.pp.024-027

Тихомирова Т. А., Федоренко Г. Т., Назаренко К. М., Назаренко Е. С.
ВЕРСАЛЬНОЕ ПРЕДСТАВЛЕНИЕ ПОЛИТОПОВ
(c. 24-27)

Аннотация. Дано описание векторного представления политопов на основе версальных моментов вершин, вычисляемых с использованием матрицы Адамара. Приведены критерии ортогональности и невырожденности базисов, симметрично присоединенных к политопам. Показано, что оба критерия могут быть удовлетворены при вложении 3D-политопа в 4D-пространство.

Ключевые слова:  политоп; матрицы Адамара; версальные моменты; форма и смещение; симметрично присоединенный базис; тональ.

Tikhomirova T. A., Fedorenko G. T., Nazarenko K. M., Nazarenko E. S.
VERSAL DESCRIPTION OF POLYTOPES
(pp. 24-27)

Abstract. An original method for representing polygonal models of geometric objects in the form of vectors of a multidimensional space with a full vector basis symmetrically attached to them is described. A distinctive feature of the method is that the representation vector is formed from the statistical moments of the coordinates of the vertices calculated using the elements of Hadamard matrix as weights. Such an approach significantly reduces the sensitivity of the polytope transform operator defined by means of attached basis to the permutation of indexes of polytope vertice, which is important in modeling complex 3D-shapes.Simple criteria for the nondegeneracy and orthonormality of a polytope’s symmetrically attached basis is presented, and a direct method for it’s orthogonalization is indicated when imbedding of a 3D-polytope into 4D-space. As a result, it is possible to considerably simplify the calculations connected with the morphism of polytope models, in particular, with the “patternbased ” transformation.

Keywords: Polytopes; Hadamard matrices; Versal moments; Shape and displacement; Symmetrically attached basis; Tonal.