[url=https://www.geogebra.org/m/ck6ecca5#material/svrgxrws]Using the tools of the P-model,[/url] we examine how the following statements are reflected in the model. How could our readers, who are acquainted with the properties of inversion (circle mirroring) substantiate the following statements?[br][list][*][color=#9900ff]Two points fit one and only one line.[br][/color][/*][*][color=#9900ff]The line divides the plane into two parts – half planes.[br][/color][/*][*][color=#9900ff]Any line in the plane defines a transformation called axial reflection, which[br][/color][/*][list][*][color=#9900ff]creates a mutually clear correspondence between the two points of the half planes;[br][/color][/*][*][color=#9900ff]original point of a mirror image of a point is on the same line;[br][/color][/*][*][color=#9900ff]mirror images of the points on a line also fall on a line;[br][/color][/*][*][color=#9900ff]has exactly one mirror axis at any two points in the plane;[br][/color][/*][*][color=#9900ff]is also an axisymmetric mirror image of a pair of axisymmetric points. [/color][br][/*][/list][/list]One possible path to the axiomatic construction of geometry is to accept the properties of axis reflection just listed as axioms. If we do this, we can say the following – absolute geometric – definitions: [br][list][*][color=#9900ff]Two lines are called perpendicular if the reflection of one on the other is itself. The perpendicularity between the lines is a symmetric relation.[br][/color][/*][*][color=#9900ff]Two sections – usually two planar geometric shapes – are regarded to be equal (congruent) if they can be transferred to each other by a series of axial reflections.[/color][br][/*][/list]As we will see, all other congruence transformations in the plane can be given by axial reflections. Furthermore, the concept of a circle(line) can be interpreted without interpreting the length (measurement) of the section. [br][list][*][color=#9900ff]Let be an O and an A point in a given plane. The outline of the circumferential point A with the centre O is called geometric position of the points A’ for which it is satisfied that OA and OA’ coincide.[/color][br][/*][/list]This structure based in axial reflection allows the development of the following concepts: [br][list][*][color=#9900ff]perpendicularity between lines;[br][/color][/*][*][color=#9900ff]perpendicular bisector (mirror points of two points);[br][/color][/*][*][color=#9900ff]angles of two half-straights;[br][/color][/*][*][color=#9900ff]congruence (equality) between angles;[br][/color][/*][*][color=#9900ff]the angle bisector (two half-straight mirror axes with a common starting point).[/color][br][/*][/list]With the help of the P-model, the milestone of the statement of the parallelism axioms is drawn more and more precisely whereby separating the concepts common to hyperbolic geometry – the absolute geometry – from the concepts valid only in the Euclidean or only in hyperbolic geometry.[br][list][*][color=#ff0000]In a hyperbolic plane, a given H-line and a non-matching H-point have at least two H-lines that fit that point and do not intersect the given H-line. [/color][br][/*][/list]This is the most important relation in which the P-model distinguishes from the well-known relations from Euclidean geometry. It can be proved – and this is also reflected in the P-model – [color=#ff0000]that there are an infinite number of lines that fit at a given point and that the given line intersects respectively does not intersect, i.e. is parallel. [/color]These two heap of lines are separated by two lines which neither intersect eachother. These are [color=#ff0000]asymptotically parallel,[/color] otherwise (in short) [color=#ff0000]unidirectional,[/color] straight lines, the other non-intersecting ones are called [color=#ff0000]ultra-parallel [/color]or [color=#ff0000]deflective.[/color] [br][br]According to this, the mutual positions of two H-lines can be[color=#9900ff] intersect,[/color] [color=#ff0000]unidirectional[/color] or [color=#ff0000]ultra-parallel. [/color]

We have just given the lines with two points each. It is relatively easy to get two lines asymptotically parallel by relocating these points. For this purpose, we used a little programming trick that we recommend to our readers who are interested in creating GeoGebra files. This is because the moving of the points is as difficult to achieve as if we want to move the two points so that their straight line fits a very far away point (possibly not even on our drawing sheet). Thus, we have already reached one of the deepest connections of the Bolyai geometry: How to edit a line that is asymptotically parallel and fits at a given point with a given line? We will return to this here.[br]For the time being, let us be satisfied with the fact that we can expand the set of points of the hyperbolic plane by the so-called infinitely distant points. Otherwise, these are also called directions. These points fit on the base circle line in the P-model. The infinitely distant points of the hyperbolic plane are represented by [color=#6aa84f] ▶[/color] in the P-model. [br][br]From the applet above you can read that[br][list][*][color=#ff0000]Each H-line has two infinitely distant points. For each line (AB) in the H-plane and its non-fitting point C there are exactly two lines which fit C and are unidirectional with (AB).[/color][br][/*][/list]