Thanks, John! I'll just add that this is Birkhoff's definition
between =
|- !a x b. between x (a,b) <=> dist(a,b) = dist(a,x) + dist(x,b)
used in Moise's & Venema's geometry books, and
The "angle" function is defined (via "vector_angle" in the file
"Multivariate/geom.ml") in a symmetrical way to be whichever of the
two angles between two vectors is non-reflex
This is Hilbert's definition, also used by Moise & Venema, and
For some purposes, it would be more useful or natural to use
directed angles.
Only Birkhoff did this, and his directed angle caused him a lot of
trouble. Birkhoff's 1932 Annals paper used an axiom saying some
function is continuous in order to deal with a directed angle problem
(the triangle angles are either all clock- or all counter-clockwise),
and it took an erratum and a paper by MacLane 20 years later to prove
this. So I think you have the right definition.
--
Best,
Bill
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