Lab 7: 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 how it could be carried out on the sphere with the tools you have available.

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).

Stereographic Planar Model of Spherical Geometry = S-geometry

In this lab, while is it important to keep the geometry on the sphere in mind at all times, the actual construction will take plane inside the Stereographic Planar Model of Spherical Geometry. One can think of this as working on a flat map of the sphere with the correspondence between sphere and plane provided by stereographic projection. So the elements of the geometry are:

A special circle E called the Equator. This circle is the considered the stereographic image of the equator on the sphere when projected from the North Pole N. The center of E is denoted S, thinking of S as the South Pole.

• S-points are the points of the plane plus an additional point at infinity.
• Antipodal map; The antipodal point Q* of an S-point Q is the S-point such that Q and Q* are stereographic images of antipodal (opposite) points P and P* on the sphere. The point Q* can actually be constructed as the inversion Q' of Q in E, followed by the point reflection of Q' in S.
• S-great circles are either Euclidean lines through S or Euclidean circles c in the plane so that the image of c by the antipodal map is c itself. It can be proved that it suffices that for a single Q in c, then Q* is also in c, then c is a great circle (and thus the antipodal points of all points in c are also in c).
• S-circles are either Euclidean lines or Euclidean circles c. Any S-great circle is an S-circle.
• S-angles between S-circles are the same as the Euclidean angles.
• Inversion in an S-circle is the usual inversion or line reflection. S-reflection is inversion, when the inverting circle is an S-great circle.
• S-distance between two S-points is the distance on the sphere of their inverse spherical images. Since distance (measured in degrees) between two points on the sphere equals the angle between the polars of the points, S-distance can also be measured as an angle between two S-great circles.

NOTE: When it is clear that we are in the plane and not on a sphere, we will say a circle or line in the plane is a "great circle" instead of an "S-great circle" just for simplicity.

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

Task 0. Building an antipodal point tool with GSP

Read the background sheet linked above. Construct the figure on the background sheet.

• Start with a circle E with center O through point R.
• Then draw a point Q and construct the rest of the figure as shown in order to construct Q*.
• In your figure, hide the lines and make a tool that constructs the Q* from Q and the givens for the circle E. (If you like, 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. Construct the circle h through A, B and the antipodal point A*.
• Check that this circle h also passes through B*. (Note: If S, A, B are collinear, the great circle will be a line.)
• Check that h intersects E in diametrically opposite points.
• Make a note explaining why must be the image of a great circle (using the properties of stereographic projection).
• Make a GreatCircle AB tool. (This will have the equatorial givens as givens in the tool as well as A and B.)
• Use your tool to draw great circles that form a triangle ABC in S-geometry. Note the triangle A*B*C*.

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

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

• This great circle g is a (spherical) diameter of the circle.
• See what the locus of g looks like as A varies. This should look familiar. You should be able to spot the two S-centers of c.

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. Hide some objects. Make a Circle Centers tool from this that constructs both centers. Note: If applied to great circles , this same tool is also a Pole Tool.

Task 4. Given an S-great circle c and an S-point A (with A not being 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 S- perpendicular bisector of A and B is the S- great circle that S-reflects A to B. This circle also reflects the antipodal point A* of A to the antipodal point B* of B. It is orthogonal to the great circle through A and B.

• Use these properties of the perpendicular bisector to construct the S-perpendicular bisector of two general S-points A and B. Make an S-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 at an S-point O as they should be?
• Construct the S-circle d through A, B, C (note: this is just the Euclidean circle through 3 points!).
• Investigate how the S-great circles through O are related to d. Can you justify the statement that O is the S-center of d?
• Show how to construct an S-circle given its S-center and an S-point on the circle. Hint: This is an old story.

Note: This is the beginning of a Construction Portfolio Problem. See the end of this page for more.

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.

Part D. Polar triangle

Task 8. Given an S-triangle ABC, study the polar triangle.

Make tools Pole and Polar. (There will be some special cases that the tools can't handle.)

• The Pole tool should take a great circle g through A and B and construct the two pole points G and G* of g.
• The Polar tool should take a point P and construct the polar circle p of P. (In other words, p is the perpendicular bisector of P and P*.)

Polars of bisectors

• Start with S-points A and B and g, the S-great circle though A and B. Construct the S-perpendicular bisector n of AB. Intersect g and m to construct the S-midpoints M and M* of AB.
• Then construct the polars a, b, m, of A, B, M, and also the poles G, G*, M, M* of the great circles g and n.
  • Are the polars concurrent? Where? Why?
  • How are the angles between the polars related? Why?
  • What is the great circle through G and M? Why?

Polar triangle of ABC

• Starting with an S-triangle ABC, construct the polar triangle.
• Measure the S-lengths of the sides of ABC by measuring angles in the polar triangle.

Relations to construction portfolio

In addition to Lenart sphere constructions, you will be asked to create some S-versions with Sketchpad

S- Figure 1. Construct an S-triangle ABC by 3 great circles and then construct the S-perpendicular bisectors of all the sides of all the 8 triangles formed the great circles. Use color or thick/thin to make this as clear as you can.

S-Figure 2. Construct the polar triangle of ABC. Then construct the interior and exterior angle bisectors of ABC. Are there additional angle bisectors of the other triangles among the 8? Hint: From your work above, you can do this by taking poles and polars of everything in a copy of Figure 1 (just make a Hide/Show button so that you can hide Figure 1).