[jacobolus]

> tan(π/8) would be halfway between 0 and 1 ie .5?

tan(π/8) = √2 - 1, not 1/2, but the value is indeed halfway between 0 and 1 if you take the distance function to be

d(a, b) = |b - a| / (√(1 + a²)√(1 + b²)) or

d(a, b) = |b - a| / |1 + ab|

(These are chordal distance or stereographic distance, respectively, when the real number line is used to represent points on a circle under stereographic projection.)

The halfway point is 0.5 when you use the distance function d(a, b) = |b - a|.