## Pseudosymmetry of General Collinear Planes

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##### 2008

##### Conference object (Published version)

,

International Society for Geometry and Graphics (ISGG)

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Show full item record##### Abstract

The paper is considering a special case of general collinear planes, which creates an
illusion of axial symmetry of the corresponding figures, for the specially chosen method of mapping. Comparing the original figure in one plane with the associated figure in another plane, without showing the constructive method of mapping, the observer would be misled to a conclusion that the type of correlation of the planes comports with the case of affine association of points, more precisely - the axial symmetry. However, correspondent points do not assort to axial symmetry, moreover, there is neither definite nor infinite collineation center in which the linker-lines would intersect. This is, actually, the case of general collinear planes, which infinitely distant points are mapped to vanishing points on a vanishing line of corresponding plane. The kind of general collinear planes that would provide such a case of pseudosymmetrical plane figures, polygons or curves, imply the vanishing lines to... form the same angles with the double straight line – the line of the planes that is mapped to itself. It is important, besides, that the axes of the mapped curves must be overlapped with the main perpendiculars of the planes. These perpendiculars are radical axes of the mapped absolute involution from the infinitely distant straight line of one plane, to a vanishing line of another. In the same manner, a circle could be mapped to an identical circle, a square to an identical square, even regular polygons can be mapped to identical and apparently symmetrical figures, as well as parallelograms and rectangles, nevertheless the planes are not affine. This result can be accomplished thanks to the special choice of the particular points, which provides the anticipated solution.

##### Keywords:

collinear planes / affinity / pseudosymmetry / vanishing lines / vanishing points

##### Source:

Proceedings of the 13th International Conference on Geometry and Graphic - ICGG 2008, 2008, 1-7##### Publisher:

- Dresden: Technische Universität Dresden; International Society for Geometry and Graphics (ISGG)