Reflection from Curved Mirrors in a Plane

L. A. Zhikharev

doi:doi:10.12737/article_5c9203adb22641.01479568

UDC 51

Reflection from Curved Mirrors in a Plane

Published в Geometry & Graphics · Volume 7, Issue 1, 2019 · Pages 46–54 · Rubrics: Scientific problems of geometry

DOI 10.12737/article_5c9203adb22641.01479568

Received: 20.03.2019 Accepted: 08.04.2019 Published: 08.04.2019 Language of publication: RUS

Authors

L. A. Zhikharev ¹

1 Moscow Aviation Institute (National Research University) (assistant)
Moskva, Moscow, Russian Federation

ABSTRACT

Reflection from a certain mirror is one of the main types of transformations in geometry. On a plane a mirror represents a straight line. When reflecting, we obtain an object, each point of which is symmetric with respect to this straight line. In this paper have been considered examples of reflection from a circle – a general case of a straight line, if the latter is defined through a circle of infinite radius. While analyzing a simple reflection and generalization of this process to the cases of such curvature of the mirror, an interesting phenomenon was found – an increase in the reflection dimension by one, that is, under reflection of a one-dimensional object from the circle, a two-dimensional curve is obtained. Thus, under reflection of a point from the circle was obtained the family of Pascal's snails. The main cases, related to reflection from a circular mirror the simplest two-dimensional objects – a segment and a circle at their various arrangement, were also considered. In these examples, the reflections are two-dimensional objects – areas of bizarre shape, bounded by sections of curves – Pascal snails. The most interesting is the reflection of two-dimensional objects on a plane, because the reflection is too informative to fit in the appropriate space. To represent the models of obtained reflections, it was proposed to move into three-dimensional space, and also developed a general algorithm allowing obtain the object reflection from the curved mirror in the space of any dimension. Threedimensional models of the reflections obtained by this algorithm have been presented. This paper reveals the prospects for further research related to transition to three-dimensional space and reflection of objects from a spherical surface (possibility to obtain four-dimensional and five-dimensional reflections), as well as studies of reflections from geometric curves in the plane, and more complex surfaces in space.

KEYWORDS

reflection curvilinear mirror dimension circle Pascal snails three-dimensional reflection model.

Information References

Authors:

1. Moscow Aviation Institute (National Research University) (assistant)

Moskva, Moscow, Russian Federation

Type:

Article

Pages:

from 46 to 54

Status:

Published

Subject area:

UDC 51

Language:

Russian, English

Keywords:

reflection, curvilinear mirror, dimension, circle, Pascal snails, three-dimensional reflection model.

References

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