N-n-digit interrelations between the sets within the R ² plane generated by quasi-rotation of R ³ space

Abstract

The background of description of the method for geometric objects rotation around the curved axes in the form of curves of the second order is being considered. The above-described method got the name of “quasi-rotation”. The algorithm of geometric construction of the two-two-digit interrelation on the basis of the circular axis of symmetry is provided. The described interrelation is analogous to the mirror symmetry regarding a straight linear axis. The method is suitable for construction of some well-known algebraic curves. The analogies between rotation and quasi-rotation are determined and described. The formula of the same plane preimage-to-image interrelation, generated by the quasi-rotation around the curved axis of the second-order curve is provided. The presented formula allows to derive the equations that effectively describe images of the given sets. The method for geometric construction of the images of geometrical objects lying within the plane of the second-order curve at their quasi-rotation for a given angle regarding the curve is described. The obtained results describe geometrical basics for the quasi-rotation. They serve as the basis for shape-forming of algebraic surfaces of high orders, and can be utilized as the theoretical basis for the computer-aided automated geometric projection.

Application of quasi-rotation surface segments in architectural prototyping

“…An algorithm of quasi-rotation surface construction based on the above reasoning can feature the initial data in the form of parameters of the desired surface, namely D and G (Figure 2) instead of elliptic axis parameters. It is important to note that the studied quasi-rotation surface is described by three parameters included in the geometric part of its determinant (1). Any of the three parameters can be replaced with one of the parameters of the desired surface specified on Figure 1.…”

“…The analytical entry of the quasi-rotation apparatus presented in paper [4] was used to create an algorithm in the environment the computer algebra system (Maple). The images of the quasi-rotation surfaces presented in papers [1][2][3] demonstrate the possibility to create surfaces of various shapes through application of the quasirotation apparatus to various generating curves. The present paper demonstrates the possibility of application of quasi-rotation surface segments in architectural form creation.…”

“…Papers [1] consider theoretical basics of the geometrical correspondence known as "quasi-rotation". Its capability to form cyclic surfaces is covered in papers [1][2][3].…”

“…Papers [1] consider theoretical basics of the geometrical correspondence known as "quasi-rotation". Its capability to form cyclic surfaces is covered in papers [1][2][3]. The analytical entry of the quasi-rotation apparatus presented in paper [4] was used to create an algorithm in the environment the computer algebra system (Maple).…”

Plane Tangent to Quasi-Rotation Surface

“…For example, the point L' constitutes the result of quasi-rotation of the point L around the axis i on 180° angle around the closest center of rotation S'; k' is the circle correspondent to this quasirotation centered at the point S'. The quasi-rotation correspondence is detailed in papers [1,2,3,4,5]. Quasi-rotation, just as regular rotation, can be applied in solution to various problems of formation.…”