Module 6

by Leigh Hollyer

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Table of Contents

Module 6

Handout by John C. Polking

We’ll use the sphere as the surface for our geometry. Consistently we will compare the relationships and definitions in our spherical geometry to planar Euclidean geometry.

Points will be the same kind of undefined object as in Euclidean Geometry

Definitions - the 1st few:






Sharing no points




Great circle

In spherical geometry, the “great circles” are not limited like the latitude lines, any antipodal points can be joined by a great circle. The lines of spherical geometry will be defined to be “great circles”.



Whether the points are

Back to spherical geometry

Exercise 1, Notebook Problem


3. In geometry, partial coincidence between two figures.

Incidence Axioms:

A3 Every two distinct lines have at least one point on them both.

Incidence Relations

Two distinct great circles meet in exactly two antipodal points.




Let’s build a table of these measurements using an “exposed” great circle carefully designed to have a radius length of one unit.. Notice that we’re actually assigning a number to an angle measure. The convention is to consider a radius co-linear with the “x axis” (zero degrees) and rotate it counterclockwise.

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Angles in radians

You may move the rest of the way around the circle using the conversion factor

I’d suggest using symmetry and arithmetic fact to assign the other measures. I’ll write it out for you.

Summarizing in spherical geometry:

As a check:


One spherical isometry is a rotation around a diameter. How would I model this?



Set up lunes on your model and notice that the vertices are antipodal points, and that the two angles are equal.

Spherical Angles

Spherical Angle

If we align two lines in this tangent plane with the two intersecting spherical lines, we can measure a planar angle (in radian measure, by definition).

We will define the measure of the angle between two spherical lines to be the same measure as the corresponding angle in the tangent plane.

Spherical Area

If you continue to divide the surface into n lunes by drawing lines through the same intersection point (evenly spaced), you’ll end up with

What if we wanted to look at some number of these lunes - what if the distance from point A on line one and point B on line two is 3p/4…and I want the area of the lune created by these lines?

The degree measurement from A to B is 135.



We’ll stick to the smallest of the regions as our convention...

Theorem 1.1.1, p.3

In spherical geometry,

It has been proven that our theorem 1.1.1 is equivalent to the parallel postulate.

Is there another way to deny the parallel postulate and what do you suppose happens to triangles when we do it?

What part of algebra is reminiscent of what we’ve got here?

Be sure to try to measure these angles on your own with your model…or use a really big beach ball (often larger models help with accuracy on this exercise)

Girard’s Theorem


I will follow the coding on your handout and label the angles of T: r, b, and g.

Using three lines has created 3 pairs of lunes. I want to use my model and sketch these lunes.

Note that each of the six lunes is composed of two triangles.

Recall that we do know the area of a lune (biangle):

We have two triangles, T and T*. One of these two shows up in each of the six lunes. The means that I have six distinct triangles plus T and T* making up the sphere.

I’ll follow the handout convention and label each lune by the letter at the “top” of the lune containing T.

We’ll use the fact that T and T* are congruent, and that the areas are the same without proof.

Summarizing our “lune list”:

Divide both sides by 4;

What does this tell you?

Notice that you are subtracting p and still have an area, a positive number.

What if we solve for the sum of the angles?


Another item to notice is that the excess:

Does the notion of similarity hold in spherical geometry?

Compare & Contrast




In spherical geometry

Concept of “betweeness” is gone.

Intersection of two lines is two points

AAS is not a congruence condition --

What about quadrilaterals?

Saccheri Quadrilaterals

He had evidently built upon the work of the Arab mathematician Omar Khayyam (c. 1050 - 1130) who also worked with this quadrilateral.

It was remarkable in that he actually worked with the quadrilaterals of spherical geometry without really recognizing what he’d done. After the framework for alternative geometry was beginning to be understood, his work was rediscovered (about 1900). In his honor, the “rectangles” of spherical geometry are called Saccheri quadrilaterals.

Generically, the shapes with two right angles are called biperpendiculars. The bottom segment is the “base”, the two sides are called “legs”, and the top is the “summit”. Base angles of a biperpendicular are right angles and equal.


Sketch of proof:

Now AD = BC why?

Notebook Problems


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