Plane Geometry - Bruce E. Shapiro

314 SECTION 53. SPHERICAL GEOMETRYWe define the distance between points as the arc length of the great circleconnecting them.Figure 53.1: There are two possible paths between A and B on any givengreat circle, with arc-lengths of a = αr and b = βr.Definition 53.7 Let A and B be points on the surface of a sphere of radiusr and center O. The the spherical distance between A and B isd(A, B) = rµ(∠AOB)where the angle is measured in radians. On the unit sphere S, we haved(A, B) = µ(∠AOB)Of course now we have the following problem: if we put any two points on acircle, there are two different paths from A to B. If we length of path fromA to B is d = d(A, B) then the distance of the other path is d ′ = 2πr − dAt times, we will be interested in each of these distances, so we cannot justlimit our angles to be smaller than π.This problem arises because betweenness fails on a great circle. Given any3 distinct points on a great circle, each of them is between the other two,although we have to be careful which of the distances we use in each case.Instead, we define a concept of separation. We say that points A and Dseparate points B and C if B is on one of the arcs from A to D and C ison the other arc (figure 53.2). Hilbert’s axioms of betweenness are replacedwith a set of axioms of separation.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012