We have already met i-lines which meet C twice, but not orthogonally - hypercircles.
We showed that they could be defined as loci. In detail, if P is a point and L is a
hyperbolic line not through P, we defined the hypercircle C(P,L) as the set of points
on the same side of L as P, and equally distant (in the hyperbolic metric) from L.
If P is on L, we simply say C(P,L) = L. If we take L to be a diameter of C, then we
get the e-lines with the ends of the diameter as boundary points.
Of course, this still uses the idea of a diameter, but this is simply a hyperbolic line
We shall refer to L as the hyperbolic line associated with the e-line.
Since our grasp on e-lines as loci is somewhat tenuous, the proofs of (1) and (2) are
To verify that the geometry of e-lines is indeed euclidean, we use some algebra.
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