Mathematical Models
Mathematics uses a variety of models in its various branches. Indeed much of applied mathematics can be thought of as designing and studying a mathematical construct that, in some sense, models the behavior of a physical or social situation. But mathematics also uses this concept of modeling internally, to represent one branch or type of mathematics within another, a means of representing the less familiar by using something more familiar.
An example familiar to every student of mathematics is found in Analytic Geometry. Each point in a (Euclidean) plane is assigned an ordered pair of real numbers as its coordinates, and then familiar geometric figures (lines, circles, parabolas, etc.) are "transferred" to (represented by) sets of pairs of real numbers that satisfy certain types of equations. Thus we come to identify the algebraic entities (equations) with the geometric entities (to such an extent that we even blur the language with phrases such as "the equation of a circle").
In the context of Lobachevskian geometry (the less familiar), a model is an assignment of mathematical objects from a more familiar area (say, Euclidean geometry) to "play the role of" the non-Euclidean objects. Several distinct models have been developed to represent the Lobachevskian plane; they can be found in the references.
The model presented in this web-site is the disk model usually attributed to Henri Poincaré: the Lobachevskian plane is represented by the interior of a disk (the set of all points inside a particular boundary circle) in the Euclidean plane. Distance between points in the (infinite) Lobachevskian plane is represented in the model by defining distance as follows:
Let the boundary circle be the unit circle centered at the origin of the complex plane. Let w and z be complex numbers corresponding to any two points interior to the unit disk; and let w' and z' be the corresponding points where the circle through w and z orthogonal to the unit circle (or the diameter through w and z) meets the boundary circle. Then the distance from w to z is:
ln( Re[ | [(w' - z) / (w' - w)] x [(z' - w) / (z' - z)] | ] )
The visual import of the disk with this metric is that a pair of points with a given distance between them will appear to be closer and closer together as their location approaches the boundary circle. Or, equivalently, a pair of points near the boundary of the disk are actually farther apart (via the metric) than a pair near the center of the disk which appear to be the same distance apart.
With this metric, the geodesic (path of minimum distance) between two points is the arc of the orthogonal circle (or the diameter) through those points. I.e., Lobachevskian lines are represented either by open-ended arcs of Euclidean circles that are orthogonal to the boundary circle or by diameters of the boundary circle (without their end-points), and Lobachevskian segments are represented by (closed) arcs of such circles (or sub-segments of a diameter). The metric also has the effect of preserving angles from the Lobachevskian plane to the model, with the angle between two arcs of circles understood in the usual way (via tangent lines at the point of intersection). And, coincidentally, the metric has the effect of representing Lobachevskian circles as Euclidean circles interior to the disk, albeit with the center of the circle "displaced" visually toward the boundary circle.