In the hyprbolic plane page, we introduced hyperbolic lines as the intersections of
disk D with i-lines orthogonal to the boundary C. In the experimental
showed some of the basic properties of hyperbolic geometry. We used our euclidean
tools to build a disk model of the hyperbolic plane.
Here, we proceed in the reverse direction, and show how hyperbolic tools can be used
to build a disk model of euclidean geometry. This is more than an academic exercise.
If we assume that euclidean geometry is consistent, i.e. free from any contradictions,
then anything built from it will also be consistent. Once we have constructed a model
of euclidean geometry with hyperbolic tools, we deduce that if hyperbolic geometry
consistent, then so is euclidean geometry. Thus the two are equally consistent.
An i-line L cuts a circle C diametrically if it meets C in diametrically opposite points.
If L cuts C diametrically at the points X,Y then these are the boundary points.
Note that the definition is asymmetric - even if L is a circle, it does not follow that C
cuts L diametrically. Also, an extended line cuts a circle diametrically if and only if
it s a diameter of the circle.
|We are now ready to build our model. We take as our set of points the set D:|z|<1,
and denote its boundary by C, as usual.
An e-line is a set of the form LnD, where L is an i-line which cuts C
Of course, the e-lines include the diameters of C, but no other euclidean segments.
The CabriJava applet allows you to experiment. You can drag X,A,X' or A' to vary
If you drag A or A' out of the disk, the associated e-line disappears.
By experimenting, you should convince yourself of the following facts
These suggest that the geometry we have produced is euclidean.
- Given any A and X, there is an e-line through A with X as a boundary point.
- Two e-lines either have the same boundary points (and do not meet in D),
or meet exactly once in D. In the former case, we say they are parallel.
- Given two points A,B, ther is exactly one e-line through A and B - to see this,
drag X round the boundary.
- Given the e-line X'Y' and the point A, there is exactly one e-line through A
parallel to X'Y'. To see this, drag X on top of X'.
We need to prove that these are theorems of our new geometry, but first, we must
deal with a basic flaw in our definitions. We used i-lines and the concept of cutting
diametrically in the definition of e-lines. These are euclidean tools. We need to
give alternative descriptions in purely hyperbolic terms.