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

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

In the isotropic plane there are seven types of conics relative to their position to the absolute figure (F, f). The paper shows how to construct an osculating circle in any point of a real conic. In the Euclidean plane every conic has at least one real vertex, i.e.a point where hyperosculating circle exists. In the isotropic plane only real ellipses and hyperbolas of the 2nd type have two real vertices. For those types of conics are the hyperosculating circles constructed as perspective collinear images of a given conic with appropiately selected elements of collineation.

Osculating Circles of Conics in the Isotropic Plane

The line-inversion and pedal transformation are defined in the quasi-hyperbolic plane and certain properties of these transformations are shown with regard to analogous transformations in the Euclidean [1, 3, 10, 12, 20], hyperbolic [4, 15, 18] isotropic [17, 19] and pseudo-Euclidean plane [5, 6, 7, 14]. As it is natural to observe class curves in the quasi-hyperbolic plane, i.e. line envelopes, the construction of a tangent point on any line of the class curve obtained by the line-inversion and pedal transformation is shown.

Line-inversion and pedal transformation in the quasi-hyperbolic plane

The Wallace-Simson line is the line containing the feet of the lines perpendicular from an arbitrary point on the circumcircle of a triangle to the sides of the triangle. The line was attributed to R. Simson by J. V. Poncelet, but today it is usually known as the Wallace-Simson line since the term was introduced by W. Wallace in 1797 and it does not actually appear in any work of Simson. Various interesting properties are known such as that the set of all of the Wallace-Simson lines for a given triangle form an envelope of a deltoid which is known as the Steiners deltoid. In this work we will treat the Wallace-Simson line and Steiners deltoid, except in the Euclidean plane, as well as in three other Cayley-Klein planes: the quasi-elliptic, the pseudo-Euclidean and the quasi-hyperbolic plane.

Wallace-Simson Line in Four Cayley-Klein Planes

All perspective collineations in a real affine plane are classified according to a constant cross-ratio and the position of the center and axis. A special attention will be given to the conditions which basic elements of perspective collineation have to fulfill in order to obtain the touch or osculation or hyperosculation of two conics. On the affine models of an isotropic and pseudo Euclidean plane the osculating circle of a conic is constructed by using perspective collineations.

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Perspective Collineation and Osculating Circle of Conic in PE-plane and I-plane

Using the facts from the theory of conics, two theorems that are analogous to the theorems in triangle geometry are proved. If the pencil of parabolas is given by three lines a, b, c, it is proved that, the vertex tangents of all the parabolas in the pencil, envelop the Steiner deltoid curve δ, and the axes of all parabolas in the same pencil envelop further deltoid curve α. Furthermore, the deltoid curves are homeothetic. It is proved that all the vertices in the same pencil of parabolas are located at the 4th degree curve. The above mentioned curves are constructed and treated by synthetic methods.

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Steiner Curve in a Pencil of Parabolas

All perspective collineations in a real projective plane are classified according to a characteristic constant. In this paper it will be shown that it is possible to construct certain types of conic pencils from a given conic by using the appropriate perspective collineation. On the affine models of an isotropic and pseudo - Euclidean plane the osculating circle of a conic is constructed by using perspective collineations.

Ivana Božić

Papers

Introduction to the Planimetry of the Quasi- Hyperbolic Plane

Perspective Collineation and Osculating Circle of Conic in PE-plane and I-plane

Steiner Curve in a Pencil of Parabolas

Classification of perspective collineations and application to a conic