Lab 6: Stereographic projection

Lab Background

NOTE: Some definitions of Stereographic projection project onto the tangent plane at the pole and other versions project onto the plane of the equator. The only difference is the size of the image. It is somewhat cleaner to project onto the equatorial plane since then the points of the equator are projected to themselves and the image of the equator is the size of a great circle on the sphere and not a scaled image.

• 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

As you do each activity, consider how the construction would look and would be carried out on the sphere.

IMPORTANT! The actual size and location figures in the plane that are projections depend in size on knowing the location and size of the projection of the equator. This means that for this stereo geometry, just as we always needed the special point O for DWEG geometry, we will always need to include the equatorial circle. Thus your tools will always have the center O and a radius point R that define this circle as part of the tool. (These can be automatched if you like.)

You can add the interior of the equatorial circle to the figure as a visual enhancement. This will be the image of the southern hemisphere. You can even make the interior not arrow selectable (from Properties).

Part A: Antipodal Point and Construction of the image of a great circle

Task 0. 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 equatorial 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.

• 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 (it should be automatic). Make a GreatCircle AB tool. (This will have the equatorial givens as givens in the tool as well as A and B.)

Part B: Orthogonal Circles, Diameters of Circles, and Centers of Circles

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

• The image of c is just any circle on the plane (also lines as a special case). Since angles are preserved, this leads to an orthogonal circle construction on the plane. This great circle is a (spherical) diameter of the circle.
• See what the locus of g looks like as A varies. This should look familiar.

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.)

• Construct two great circle diameters as in Task 2 and intersect them. Make a center tool from this.

Task 4. Given a great circle c on the sphere and a point A on the sphere, not one of the poles of c, construct the great circle h through A that is orthogonal to c.

• There are a couple of ways you can do this.

Part C. Circumcircle of a spherical triangle

Task 5. Construct the spherical perpendicular bisector of A and B

On the sphere, the perpendicular bisector of A and B is the great circle that reflects A to B. This circle also reflects the antipodal point of A to the antipodal point of B and is orthogonal to the great circle through A and B.

• Use these properties of the perpendicular bisector to construct the stereographic image of the spherical perpendicular bisector of two general points A and B. Make a spherical perpendicular bisector tool.

Task 6. Given a spherical triangle ABC, construct the perpendicular bisectors of the sides, the circumcenters and the circumcircle.

• Are your perpendicular bisectors concurrent as they should be?
• How is the spherical circumcircle related to the spherical circumcenters?

Task 7. Give experimental evidence for whether or not the medians of a spherical triangle are concurrent.

Working with the previous figure, the perpendicular bisectors intersect the sides at the spherical midpoints. Connect each vertex with the opposite midpoint by a great circle to construct the spherical median.

• Check whether or not your spherical medians seem to be concurrent. Note the answer for later.

Looking Ahead

If you finish the earlier work with time left, you may wish to get a head start on constructing a Wulff net. This will be a homework problem soon.

Problem 6.1 from Assignment 6: 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.