Do each activity on the physical (Lenart) sphere and also with Sketchpad on the stereographic map of the sphere. An S-point is a point on the Sphere. In sketchpad there is given a circle e with center S passing through R.
Two S-points determine a unique great circle (unless the points are an antipodal pair). Given two S-points A and B construct a great circle through A and B. (In other words, construct a Euclidean circle through A, B and the antipodal point of A. In this Sketchpad exercise you can leave off the special case when the great circle is a Euclidean line. Make a script gcAB that takes as givens, the points A, B and E and R..
There is a unique great circle through a given S-point that is orthogonal to a given S-circle m (unless the point is a center of the circle). In a sketch construct a circle m with (Euclidean) center A through B and a point C, construct a great circle n through C that is orthogonal to m. (Note two possible cases: C is on m and C is not on m. Can you handle both with one sketch?) Make a script Sperp that takes as givens the points A, B, C and E and R. [Note: This great circle is one diameter of m. In the case that m is a great circle, this is the construction of a perpendicular "line".
Given a circle c, construct the S-center of c using the diameters from 2.
Given S-points A and and B, construct an S-circle with S-center A through B. Then also construct the S-circle with S-center B through A. Let C be one of the intersections points of the two circles. Now construct the 3 great circles AB, BC, CA and measure the angles at the vertices. Are the angles equal? Do the angles stay the same when you move A and B?
Imagine a globe with parallels of latitude and meridians of longitude spaced at 15-degree intervals. Reference for nets is on this page.
Additional Links
http://www.math.washington.edu/~king/coursedir/m497w01/as/as03.html
Online books linked from Peter Doyle