On One Property of a Circle on the Coordinate Plane
| Autor: | O. Grafskiy, Yu. Ponomarchuk |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
020301 aerospace & aeronautics
Geometry 02 engineering and technology 010402 general chemistry 01 natural sciences 0104 chemical sciences Generalised circle Great circle Unit circle 0203 mechanical engineering Concyclic points Horizontal coordinate system Locus (mathematics) Fundamental plane (spherical coordinates) Circle of a sphere Mathematics |
| Zdroj: | Geometry & Graphics. 5:13-24 |
| ISSN: | 2308-4898 |
| DOI: | 10.12737/article_5953f2af770c35.65774157 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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