Where does the hyperbolic metric come from? In hyperbolic geometry, the metric is often defined as $$ds=\frac{\sqrt{dx^2+dy^2}}{y}$$ Where did this metric come from? I have thought long and hard about this question, but have no satisfactory answer. 
 A: This is the metric that results in the Poincaré half-plane model of the hyperbolic plane.
A: Notation is always troubling. The concrete calculations that go with the metric definition give what are called "geodesics," which are the replacement for straight lines, as they are the shortest curves between two points. in this case, there are just two types of geodesics, here parametrized by arc-length, vertical rays
$$ (A, e^t)  $$ and semicircles with center on the $x$-axis
$$  (A + B \tanh t, \; \; B \operatorname{sech} t )  $$
with real constants $A$ of any sign and then $B>0.$
A: As you know, the fifth euclidean postulate says: Given a straight line $L$ in the plane and a point $p$ not in $L,$ there is exactly one straight line $S$ parallel to $L$ and passing by $p.$  Changing this postulate to "there are at least two lines such that..."  leads to the so-called hyperbolic geometry. The upper-half space, $\mathbb{R}^{2}_+,$ equipped with that abstract fundamental form, or metric, is one example of a hyperbolic space, where this new "fifth postulate" holds. So, that metric comes from the attempt to construct a space with such modification of the euclidean postulate.
