Table of Contents
Handout by John C. Polking
Well 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
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
Whether the points are
Back to spherical geometry
Exercise 1, Notebook Problem
3. In geometry, partial coincidence between two
A3 Every two distinct lines have at least one point
on them both.
Two distinct great circles meet in exactly two
Lets build a table of these measurements
using an exposed great circle carefully designed to have a radius length of
one unit.. Notice that were 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.
Angles in radians
You may move the rest of the way around the circle
using the conversion factor
Id suggest using symmetry and arithmetic fact
to assign the other measures. Ill 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.
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
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.
If you continue to divide the surface into n lunes
by drawing lines through the same intersection point (evenly spaced), youll end up
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
and I want the area of the lune created by these lines?
The degree measurement from A to B is 135.
Well stick to the smallest of the regions as
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
weve 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)
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
Recall that we do know the area of a lune
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.
Ill follow the handout convention and label
each lune by the letter at the top of the lune containing T.
Well 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
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?
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 hed 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?