Lab 4: Stereographic projection

Lab Background

• Background sheet

More of Stereographic projection references

• Wilson Strothers' pages on Stereographic Projection and Inversion
• Stereographic Projection defined on Mathworld

Lab Activity: Geometry on the Sphere

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.

Making an antipodal point tool with GSP

Construct the figure on the background sheet. Start with a circle E with center O through point R. The draw a point Q and construct the rest of the figure as shown. Hide the lines and make a tool that constructs Q* from Q and the givens for the circle. (If you know how, you can make the tool automatch the circle center O and radius point R.

Task 1. Construct a great circle through 2 points

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.

• With Lenart sphere, draw two points and use the ruler (great circle tool).
• With GSP, draw two points A and B. The image of the great circle in the plane will be a circle in the plane (or possibly a line in special cases) through A, B and the antipodal point A*. Construct this circle. Check that the circle also passes through B* as it should. Make a GCircle AB tool.

Task 2. Given a circle c on the sphere and a point A on c, construct a great circle g through A that is orthogonal to c.

Recall that the center of the

• With Lenart sphere, use right angle on the ruler.
• With GSP, to construct the stereographic image g' in the plane, recall that the center of g' will have to be on the line through S tangent to c. Also the center will have to be on the perpendicular bisector of A' and A*'.

Task 3. Given a circle c on the sphere and a point A on c, construct the centers K and L of the circle c. (If c is a great circle, the centers are the poles of c.)

In each case, construct two great circles as in Task 2 and intersect them.

Task 4. Given a great circle c on the sphere and a point A on the sphere, construct the great circle h through A that is orthogonal to c.

Method: If K and L are the poles of c, then the circle h will have to pass through A, K and L.

Task 5. Wulff Nets and other nets

Imagine a globe with parallels of latitude and meridians of longitude spaced at 15-degree intervals. Reference for nets is on this page.

• Construct the image of the southern hemisphere when projected from the north pole.
• Construct the Wulff net, which is the projection of a hemisphere cut-by a north-south meridian great circle projected with center somewhere on the equator.