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Singular Frégier conics in non-Euclidean geometry. (English) Zbl 1421.51007
Author’s abstract: “The hypotenuses of all right triangles inscribed into a fixed conic \(\mathcal{C}\) with fixed right-angle vertex \(p\) are incident with the Frégier point \(f\) to \(p\) and \(\mathcal{C}\). As p varies on the conic, the locus of the Frégier point is, in general, a conic as well. We study conics \(\mathcal{C}\) whose Frégier locus is singular in Euclidean, elliptic and hyperbolic geometry. The richest variety of conics with this property is obtained in hyperbolic plane while in elliptic geometry only three families of conics have a singular Frégier locus.”
Frégier points and Frégier conics.
There are different generalizations of the theorem of Thales. One of them is the Frégier Point of a conic.
To find this point, pick any point \(P\) on a (non-degenerate) conic, and draw a series of right angles having \(P\) as common vertex. The rays of the right angles intersect the conic again and the line segments that connect the intersection points always concur in a point \(P'\), the Frégier point.
It is easy to construct \(P'\) using just two lines, and \(P'\) can be used to construct a tangent to a conic on a point \(P\) on this conic.
If the point \(P\) itself moves on the conic, the locus of the Frégier points moves along a conic section with the same centre as the original conic.
Frégier points and conics in non-Euclidean geometry.
This note proves twice, that the Frégier points and Frégier conics exist also in the non-Euclidean plane. (An important tool is to use a projection that preserves angles at \(P\).)
Using homogeneous coordinates the notes calculates then the equation of the Frégier conic of a conic \(\mathcal{C}\) in paraxial position.
It discusses furthermore the question when the Frégier conic is singular, i.e. when it is a degenerated conic, because in this case the Euclidean and the non-Euclidean geometry differs there. The coordinates of these singular Frégier conics is given together with the geometric reason why they are singular. (Of course the hyperbolic plane admits the greatest number of different cases: The singular Frégier conic might be a circle, a parabola, an osculating parabola, a horocycle or even singular in general.)
51M09 Elementary problems in hyperbolic and elliptic geometries
51M04 Elementary problems in Euclidean geometries
51N35 Questions of classical algebraic geometry
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