__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:
Sphere
Aside:
Center
Radius
Aside:
Sharing no points
Aside:
Tangent
Antipodal
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”.
Aside:
Geodesics
Whether the points are
Back to spherical geometry
Exercise 1, Notebook Problem
Incidence
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.
Definition
Distance
Aside
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:
Isometry
One spherical isometry is a rotation around a
diameter. How would I model this?
Shapes
Lunes
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.
Generalizing
Triangles
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
Preliminaries:
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?
Consequences
Another item to notice is that the excess:
Does the notion of similarity hold in spherical
geometry?
Compare & Contrast
Same:
Aside
Differences:
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.
Theorem
Sketch of proof:
Now AD = BC why?
Notebook Problems
Generalization
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