DOI: 10.14489/vkit.2017.06.pp.017-024

Тихомирова Т. А., Надыкто А. Б., Назаренко К. М.
ТОНАЛИ: АЛГЕБРА И ГЕОМЕТРИЯ САМОПОДОБИЯ
(c. 17-24)

Аннотация. Введено понятие тоналя – универсального базиса, присоединяемого к вектору размерности 2n на основе соотношений перестановочной симметрии его компонент. Даны описания структуры и свойств тоналей. Показана их  связь с комплексными числами, кватернионами и октавами. Рассмотрены принципы применения тоналей для представления отношений геометрического самоподобия при интерполяции и скейлинге полигональных моделей кривых, поверхностей и тел, вложенных в многомерные пространства.

Ключевые слова: векторная алгебра; многомерные пространства; самоподобие; полигональные модели поверхностей; фрактальная интерполяция; текстура; скейлинг. 

Tikhomirova T. A., Nadykto A. B., Nazarenko K. M.
TONALS: ALGEBRA AND GEOMETRY OF SELF SIMILARITY
(pp. 17-24)

Abstract. A new computationally efficient mathematical framework for self-similarity based (fractal) interpolation and scaling of polygonal meshes has been developed. A new concept of a tonal as a universal basis that can be symmetrically attached to an arbitrary vector of dimension 2n has been introduced. A tonal basis consists of vectors created by pairwise permutations of components of the reference vector. In low dimensional spaces (n < 4) tonals are equivalent to the matrix representation of complex numbers, quaternions and octaves. At n ≥ 3 a distinctive feature of tonals is the presence of side diagonals (overtones) in their cross-correlation matrix. The overtones are arranged with a pitch of 8 elements relative to the main diagonal, and their amplitudes are equal to autocorrelation of the reference vector with a cyclic shift by an integer number of octaves. Such a structure of overtones provides the two key advantages of tonals in comparison with other types of attached bases. First of all, it essentially simplifies the algorithm for tonal “inversion”, and hence, calculations of similarity relations for vectors in high dimensional spaces. Secondly, it yields to simple and efficient representation of graphic primitives, such as polygons and polytopes, initially defined in low dimensional space (n ≤ 2), as a single vector of a larger dimension space, so that the amplitudes of the overtones on the regular meshes (square, cubic, etc.) would be minimal. It has been shown that such a representation provides a greater stability of fractal interpolation and scaling algorithms compared to earlier ones. Several examples of fractal synthesis of curves and surfaces from low order polygonal meshes have been presented to illustrate the key advantages of the developed mathematical framework and concept of tonals.

Keywords: Vector algebra; Multidimensional spaces; Self-similarity; Polygonal surface models; Fractal interpolation; Texture; Scaling.