Background of Stereographic Projection

The definition of Stereographic Projection from a point N on a sphere is given in the web references.

The properties of Stereographic Projection that we will use (and later prove) are.

• Stereographic projection is a 1-1 map of the sphere onto the plane, except that projection image of the projection center N is not defined (or may be set = infinity).
• Circles on the sphere not through N map to circles on the plane.  Circles through N map to lines on the plane.
• Angles are preserved by stereographic projection.

The main construction we will use is this one.  It gives a side view or cross-section that can be used to give distances in the plane. 

Make a script for inversion (which you have) and a script for the antipodal point of P (in the plane map) which is simple the rotation of P' by 180 degrees with center O.

[If you want to make the scripts from this figure, you can hide the construction lines so that you have a script that starts with a circle OR and point P, then constructs line OP, then line NS as a perpendicular and finally point P' and/or P'' as shown.] 

How to work with Sketchpad

In a sketch, you will need to start with a circle e which you will consider the image of the "equator" which is the great circle cut by the perpendicular bisector plane of NS.

• Then for example, to construct the plane image of a great circle through points P and Q, use your script to construct plane image of the antipodal point P'' of P and then construct the circle through these 3 points. Note that the point P'' depends on e as well as P.
• Notice that this circle will automatically intersect the circle 3 in opposite points on the circle.
• To construct a circle of latitude through a point P, given two great circles of longitude c1 and c2, construct the circle through P orthogonal to c1 and c2.

How to work with the Lenart Spheres

With the Lenart spheres, you will have a "ruler" which is a great circle marked with 360 degrees. 

• The "ruler" also has a "T-square" built in for 90-degree angles. 
• You can also find the "equator" or polar circle of a point P on the sphere using this tool. 
• For spherical circles, you will need either string or ribbon or possibly a spherical compass is one is at hand.