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Geometry & Graphics
Journals / Geometry & Graphics / Volume 5 Issue 2 / On One Property of a Circle on the Coordinate Plane
UDC 51

On One Property of a Circle on the Coordinate Plane

Published в Geometry & Graphics · Volume 5, Issue 2, 2017 · Pages 13–24 · Rubrics: Scientific problems of geometry
DOI 10.12737/article_5953f2af770c35.65774157
Received: 28.06.2017 Accepted: 04.07.2017 Published: 04.07.2017 Language of publication: RUS
Authors
O. Grafskiy 1 · Yu. Ponomarchuk 2
1 Far Eastern State Transport University (Professor)
Habarovsk, Khabarovsk, Russian Federation
2 Far Eastern State Transport University (Head of Chair)
Habarovsk, Khabarovsk, Russian Federation
ABSTRACT
Descartes’ and Fermat's method allowed to define many geometrical forms, including circles, on the coordinate plane by means of the arithmetic equations and to make necessary analytical operations in order to solve many problems of theoretical and applied research in various scientific areas, for example. However, the equations of a circle and other conics in the majority of research topics are used in the subsequent analysis of applied problems, or for analytical confirmation of constructive solutions in geometrical research, according to Russian geometrician G. Monge and others, including. It is natural to consider a circle as a locus of points, equidistant from a given point — a center of the circle, with a constant distance R. There is another definition of a circle: a set of points from which a given segment is visible under constant directed angle. Besides, a circle is accepted to model the Euclid plane in the known scheme of non-Euclidean geometry of Cayley-Klein, it is the absolute which was given by A. Cayley for the first time in his memoirs. It is possible to list various applications of this geometrical form, especially for harmonism definition of the corresponding points, where the diametral opposite points of a circle are accepted as basic, and also for construction of involutive compliances. The construction of tangents to a circle can be considered as a classical example. Their constructive definition is simple, but also constructions on the basis of known projective geometry postulates are possible (a hexagon when modeling a series of the second order, Pascal's lines). These postulates can be applied to construction of tangents to a circle (to an ellipse and hyperboles to determination of imaginary points of intersection of a circle and a line. This paper considers the construction of tangents to a circle without the use of arches of auxiliary circles, which was applied in order to determine the imaginary points of intersection of a circle and a line (an axis of coordinates). Besides, various dependences of parameter p2, which is equal to the product of the values of the intersection points’ coordinates of a circle and coordinate axes, are analytically determined.
KEYWORDS
a circle on the coordinate plane; real and imaginary points; analytical and constructive solution; mathematical package Maple.
Information Text References
Authors:
1.  Far Eastern State Transport University (Professor)

Habarovsk, Khabarovsk, Russian Federation
2.  Far Eastern State Transport University (Head of Chair)

Habarovsk, Khabarovsk, Russian Federation
Type:
Article
Pages:
from 13 to 24
Status:
Published
Subject area:
UDC 51
Language:
Russian, English
Keywords:
a circle on the coordinate plane; real and imaginary points; analytical and constructive solution; mathematical package Maple.
Text

Метод Декарта и Ферма позволил в координатной плоскости числено определять при помощи уравнений многие геометрические формы, в том числе и окружности, производить необходимые аналитические операции при решении поставленных задач для многих теоретических и прикладных исследований различных научных направлений, например [1; 2; 21; 27].

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