We have already indiated that there is a concept of distance in hyperbolic geometry, and that this is related to the function D.
Definition
In the hyperbolic circles pages, we introduced the function D.
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For z, w in D0 D(z,w) = |z-w|/|w*z-1|
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Definition For z, w in D0, the hyperbolic distance between z and w given by d(z,w) = 2arctanh(D(z,w)).
Note that arctanh is an increasing bijection from [0,1) to [0,∞).
To verify that it satisfies D4, we begin by looking at the case
Lemma 1
Lemma 2 Together, these establish D4, so d is a distance function on the disk.
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The hyperbolic radius of a hyperbolic circle.
The h-circle K(w,r) consists of points z with D(z,w) = r.
The CabriJava window shows an h-segment AB with D(A,B) = 1/2.
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