Mathematics 138
Measuring Angles and Distances in Three Different Ways
We want to measure the angle between the x axis and the line y = mx.  This is defined as twice the area of the region between the two lines bounded by

the circular arc  if we want elliptic measure.
the straight line x = 1 if we want parabolic measure
the hyperbolic arc  if we want hyperbolic measure.
To duck the issue of "signed" vs. "unsigned" we assume m>0.  Then everything will come out positive.
In each case we can calculate the area easily by using polar coordinates, since twice the area swept out by a ray whose length is r(q) is given by
2A = 
Case 1: The circle.  Here r(q) = 1 and
2A = = q = arc tan m,  or m = tan (2A)
Case 2: The straight line.  Here r(q) = sec q and
2A = = tan q = m   or m = 2A
Case 3: The hyperbola. Here 
Proof: 

So 2A =  
But = = = 

or = = = 
So finally, 2A =   or     or .
Solve for m: 

Now we also have three alternative ways of measuring lengths along the vertical line x = 1. Simply join all the points on the line to the origin, and define the length of a segment to be the angle that it subtends at the origin!

Our definition of angle in terms of area then guarantees that AB + BC = AC, since
angle ab + angle bc = angle ac.
For defining length, the obvious measure of angle to use is the parabolic one.  Then the length of the segment AB is equal to the parabolic measure of the angle between lines a and b, mb - ma.  The entire line appears infinitely long, and its parabolic length is infinite.

Another possibility is to use the elliptic measure of angle.  In that case the distance between A and B is arctan mb - arctan ma.  The length of the entire line is therefore p.  This would be appropriate if, for example, you made a flat map of half the surface of the Earth by placing a sheet of paper tangent to a globe at the point where the prime meridian meets the Equator.  The prime meridian would have to be represented by a vertical line of infinite length.  On such a map, ordinary Euclidean length would give a very misleading view of the true distance between points on the prime meridian, especially near the poles, but the elliptic measure of distance, equivalent to marking latitude on the prime meridian, would provide a measure of distance that is preserved by rotations of the Earth!
In this case the line appears infinitely long when represented in a Euclidean plane, but its length is in fact finite.

The final possibility is to use the hyperbolic measure of angle.  This can only work if the vertical segment extends from -1 to +1, since if the slope m of a line exceeds (or even equals) 1 in magnitude, then the formula for angle involves the logarithm of a non-positive quantity.
So in this case the line appears finite when represented in the Euclidean plane, but its length is actually infinite.  This would be appropriate if the Euclidean line segment represented velocities in the interval (-c, c) (c is the speed of light).  If the distance between two "points" on this line is required to equal the logarithm of the Doppler shift for signals sent between two spaceships moving at different speeds, than there is no upper limit to distance.  The motions that preserve this "distance" are Lorentz transformations.

Nineteenth-century geometers, who did not know the theory of relativity, invented another way of introducing hyperbolic measurement of angles and distances in terms of "cross-ratio."  In fact, this is a little bit more general than what was done above.

Consider four lines that meet at O.  It is useful to think of i and j as being "fixed" in the same way as the speed of light is fixed; they might, for example, be specified by eigenvectors of a
linear transformation of the plane.  Lines a and b are "variable" but we want to define an angle between them that is preserved by motions that carry lines i and j into themselves (while perhaps carrying I into I' and J into J').
The way to do this is to pair a with i and with j, take the ratio of sines of Euclidean angles, then do the same for b, then take the ratio of the ratios. In the next formulas, sin (ai) will mean the sine of the unsigned angle between a and i. Cross-ratios can be defined with signed angles, but in this case the cross-ratio CR has to come out positive so keeping track of signs is unnecessary.

The cross-ratio CR(a, b ; i, j) = =
The hyperbolic measure of the angle between a and b is then ln(CR(a,b; i, j))

The cross-ratio of four points on a line can similarly be defined as
CR(A, B, I, J) = =
The hyperbolic distance between A and B is then ln (CR(A, B; I, J))
Remarkably, the hyperbolic angle between a and b equals the hyperbolic distance between A and B (or between A' and B', for that matter).  By the Law of Sines,
 ,    ,,  
Take the product of the respective sides of the first two and divide by the second two:
==1,
 since OAI and OAJ sum to p and have equal sines, as do OBJ and OBI.
Thus

In the case considered earlier,

CR(A, B; I, J) = 

CR(A, B; I, J) = 
so the hyperbolic distance AB and the
hyperbolic angle between a and b are both