Introduction to Geometry. By H. S. M. Coxeter. Pp. xiv, 443. 1961. (John Wiley)

New invariants in the family of 3-periodics in an Elliptic Billiard (EB) were introduced in [20], stemming from both experimental and theoretical work. These include relationships between radii, angles and areas of triangular members of the family, as well as a special stationary circle. Here we present the proofs promised there as well as a few new related facts.

New Properties of Triangular Orbits in Elliptic Billiards

This article provides a simple pictorial introduction to universal hyperbolic geometry. We explain how to understand the subject using only elementary projective geometry, augmented by a distinguished circle. This provides a completely algebraic framework for hyperbolic geometry, valid over the rational numbers (and indeed any field not of characteristic two), and gives us many new and beautiful theorems. These results are accurately illustrated with colour diagrams, and the reader is invited to check them with ruler constructions and measurements.

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Universal Hyperbolic Geometry II: A pictorial overview

In the Euclidean plane there are several well-known methods of constructing an osculating (Euclidean) circle to a conic. We show that at least one of these methods can be “translated” into a construction scheme of finding the osculating non-Euclidean circle to a given conic in a hyperbolic or elliptic plane. As an example we will deal with the Klein-model of these non-Euclidean planes, as the projective geometric point of view is common to the Euclidean as well as to the non-Euclidean cases.

Osculating Circles of Conics in Cayley-Klein Planes

The quasi-hyperbolic plane is one of nine projective-metric planes where the absolute figure is the ordered triple { j1, j2, F}, consisting of a pair of real lines j1 and j2 through the real point F. In this paper some basic geometric notions of the quasi-hyperbolic plane are introduced. Also the classification of qh-conics in the quasi-hyperbolic plane with respect to their position to the absolute figure is given. The notions concerning the qh-conic are introduced and some selected constructions for qh-conics are presented.

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Introduction to the Planimetry of the Quasi- Hyperbolic Plane

There are many varieties of Butterfly Theorem in the Euclidean plane. Here some analogous theorems in the hyperbolic plane are proved. It is shown that proofs do not depend on the class of circle into which complete quadrangle is inscribed. The only difference is between the case when qadrangle is inscribed into absolute conic and the case when it is inscribed into one of three classes of circles. The construction of the points with butterfly property is given. It is proved that with any quadrangle an infinite number of butterfly points is associated which are located on a second order curve.

The Butterfly Theorems in the Hyperbolic Plane

If each intersection point of a third order curve with the absolute conic of the hyperbolic plane is a tangential point, this curve will be called an entirely circular cubic. According to this definition a rough classification of such curves is given into four main types and nine sub-types. Some of them are constructed by a (1,2) or (1,1) mapping and the others are constructed by the generalized quadratic hyperbolic inversion. Thus we extend and complete Palman's paper [5] in a synthetic way.

A classification and construction of entirely circular cubics in the hyperbolic plane

The article observes a one-parameter triangle family T. We prove that the sets of the orthocenters, centroids, circumcenters and the midpoints of the variable triangle side of the triangle family T lie on four different hyperbolae. Furthermore, there is constructed an envelope $k^4_3 of the perpendicular bisectors of the variable triangle sides. Also it is constructed a bicircular rational quartic as an envelope of the circumcircles of the triangle family T.

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Family of triangles and related curves

We construct pedal curves of conics in the projective model of the pseudo-Euclidean plane (further in text: PE-plane). Generally pedal curves are circular quartics, but in certain cases they are degenerated into circular cubics or even conics, as in the Euclidean plane. Since the absolute points are real and since there are more types of conics in the PE-plane, there are many types of pedal curves of conics that we can not derive in the Euclidean plane. Those are entirely circular quartics and cubics with different types of singularities in the absolute points or special cases when a pedal curve is degenerated into such type of a conic which does not exist in the Euclidean plane. In this article we show only cases specific to the PE-plane, so we do not construct cases that are analogous to the Euclidean plane.

Ana Sliepčević

Papers

Introduction to the Planimetry of the Quasi- Hyperbolic Plane

The Butterfly Theorems in the Hyperbolic Plane

A classification and construction of entirely circular cubics in the hyperbolic plane

Family of triangles and related curves

Pedal curves of conics in pseudo-euclidean plane