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Vol 5 No 4-5
Pages:
388-394
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RUS
Research article
Articles

The theory of axial collineations as an extension of Steiner's theory of conoprimes (Kegelsсhnittbüschel)

Authors:
E. S. Fedorov
Date submitted:
1915-06-23
Date accepted:
1915-08-07
Date published:
1915-12-01

Abstract

According to Shteiner’s famous theory, two given involutions of pairs of points on straight lines on a plane determine the involution on any straight line on the plane, that is, a full second of involution. The determining factor of all these involutions is the linear prime of curves, namely conoprime (Kegelschnittbüschel according to Steiner), having common two pairs points, of which not only one, but both can be imaginary. Each straight line intersects each prime curve at a pair of points of the involution belonging to it. Specifically, we can define collineations by two axes without any involutions. If we call the axes whose points are the real double points of all involutions real, and call the axes of isotropic involutions imaginary, then we obtain that any axial collineation can be defined by a pair of axes, real or imaginary (see article).

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