The Poincaré’s hyperbolic plane projection is a representation of the complete hyperbolic plane constrained inside a single euclidean circle.
The hyperbolic plane is an impressive achievement of mathematics trying to answer the so called “parallels problem”: given a line and an external point to it, is there just one possible parallel through this point?
The answer is yes… and no. In fact there are three different possible and mathematically coherent answers: the euclidean, hyperbolic and spheric plane. The euclidean plane is the one we all learned at school. The spheric plane is easy to imagine, as it can be represented by the surface of a sphere. But there’s no isometric immersion of the hyperbolic plane in the euclidean space, so it is not easy at all to imagine it.
One representation of the hyperbolic plane is that of the Poincaré’s projection. With it, hyperbolic lines are the portion of orthogonal circles (circles that form 90º angles on the points of crossing) to that of the boundary, that fall inside the representation:
Once we know this a question arises: would it be correct then to geometrically obtain the midpoint (E) between two points (say A and B) of a line with this simple ruler-and-compass euclidean construction? :
The problem can be reconverted from the Poincaré projection to an euclidean geometry problem: can we trace a circumference orthogonal to other two and that passes trough two given points? The two circumferences would be the Poincaré projection boundary and the line on which we’re selecting two points. The latter would be an euclidean circumference orthogonal to the boundary. And the two points would be the points C and D (following notation of previous figure) in which two circumferences with centre on A and B and radius AB (but any other radius value is possible) cross.
Well it seems that it is feasible, so if we draw an example, we should find the answer. Using the concept of circle power, there exist geometric operations for finding the circumference orthogonal to two given ones which also passes trough a point (say C for example)… So if this circumference also (magically) passes through the second point D, the answer would be: yes! the construction is possible and it has different interpretations in the euclidean plane (circumference orthogonal to other two that also passes trough two given points) and in the hyperbolic plane (midpoint between two points of a line).
Well… the answer is that this geometric construction is not possible in general…
For example here the bigger black circumference with an inner point pattern would be the Poincaré boundary and the smaller one (below on the right) one hyperbolic line: in fact the hyperbolic line is the inner segment between C and D. The chosen points on that line are F and G (red coloured), and we will obtain the hyperbolic lines (pink circumferences) that would pass through H or I (the points of intersection of the circles with same diameter and center on F and G)…
…and they’re different ones, not the same at all.Though close enough so that an imperfect drawing by hand would not settle the question :-)
Here is the GeoGebra file of this construction: circumferences orthogonal to two circumferences and a point (H or I).ggb
So, why is it not possible?
The answer is that the Poincaré projection is not an isometric projection. So distances follow formulas related to hyperbolic functions, and cannot be measured with a simple euclidean ruler. In fact the distance grows more and more as we left the centre of the representation. This has the consequence that the euclidean construction to obtain the midpoint is possible… but the centre of the hyperbolic circles is not the centre of the euclidean circles we previously drew!
We can make the correct construction using this tool and obtain the awaited answer… which has no euclidean geometric counterpart contrary to what we first thought:
Here is the file of this construction: perpendicular to a segment throu its middle point.euc
Note that C is the centre of the yellow circle, and D is the centre of the green one. The midpoint between C and D is E, and the pink line who crosses through F and G, intersections of the circles, crosses the original line on the midpoint (E) as expected.
The distances CD, CF, CG, DF, DG are all the same: you can measure then with the tool used for the representation. Note also that E is not at the same euclidean distance from C and D.
That’s all because of the not isometry of the projection: distances are greater as we approach to the boundary of the projection. In fact on the boundary (which do not belong to the projection) the distance would be “infinite”. This is what makes these hyperbolic lines in fact infinite length lines (like the euclidean ones) although they’re represented as finite circumference segments.
One of its symmetric branches (except for the generatrix) corresponds to this coloured region of the hyperbolic plane using the Poincaré projection:
Note also that the pseudosphere allows us to see with our bare eyes a portion in 3D of an horocycle, the red circle in the previous figure. Incredible!
So our previous construction would succeed on this strange Minding surface… The problem is to draw lines and circles on a so damned constant curved surface :-)