Types of Affine Transformations and Their Composition

O. Grafskiy

doi:doi:10.12737/21529

Types of Affine Transformations and Their Composition

Published в Geometry & Graphics · Volume 4, Issue 3, 2016 · Pages 11–16 · Rubrics: Scientific problems of geometry

DOI 10.12737/21529

Received: 19.09.2016 Accepted: 19.09.2016 Published: 19.09.2016 Language of publication: RUS

Authors

O. Grafskiy ¹

1 Far Eastern State Transport University (Professor)
Habarovsk, Khabarovsk, Russian Federation

ABSTRACT

Known types of affine transformations which can be received as the resulting transformations when performing a combination as works of two other transformations are considered. Possibility of creation of the considered combinations as in the constructive way, and analytically on the removed dependences of functions of transformation is investigated. "The affine and projective geometry" by means of graphic AutoCAD packages, the COMPASS, or Microsoft Office Visio, and disciplines "Computing geometry" with application of a mathematical Maple package in which combinations of geometrical transformations are carried out by means of multiplication of matrixes is planned to use the received results in educational process when performing settlement and graphic works as students on discipline. These disciplines are studied by students in the direction 09.03.01 – Informatics and computer facilities (a profile: Systems of the automated design) and to the direction 09.03.03 – Applied informatics (a profile: Applied informatics in design). Field of activity of affine geometry from a position of the theory of groups F. Klein respectively are subgroups of affine transformations. In known sources studying of affine transformations from perspective and affine compliances is offered (relationship), compression (stretching), slanting symmetry, shift and their compositions (multiplication, work) are considered further. However, considering relationship on the plane, it is possible to notice that the coefficient of this transformation is negative size. Therefore, at this transformation there is an axial symmetry. Therefore, the purpose of the real work is definitions of such compositions at which performance there is a possibility of modeling of certain known affine transformations as resultants. We will in passing cite: "We will notice synthesis and the analysis, not in mathematical, and in general-logical sense of the word are absolutely equal, and in any research they constantly intertwine with each other; therefore, there can hardly be a speech about providing domination to one of these tools of human thought" (from the minutes of Physical and mathematical society of 31.03.1898, Kazan). In this regard, the received combinations should be considered and from analytical positions which provide modeling with application of information technologies.

KEYWORDS

compositions of affine transformations functions of transformations transformation matrixes visualization in Visio a mathematical Maple package.

Information References

Authors:

1. Far Eastern State Transport University (Professor)

Habarovsk, Khabarovsk, Russian Federation

Type:

Article

Pages:

from 11 to 16

Status:

Published

Language:

Russian

Keywords:

compositions of affine transformations, functions of transformations, transformation matrixes, visualization in Visio, a mathematical Maple package.

References

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