Spherical Geometry

The geometry we’ve been studying is called Euclidean Geometry The geometry we’ve been studying is called Euclidean Geometry. That’s because there was this guy - Euclid.

Euclid assumed 5 basic postulates. Remember that a postulate is something we accept as true - it doesn’t have to be proven.

One of those postulates states: Through any point not on a line, there is exactly one line through it that is parallel to the line. Number 3 on your notes! Try to draw this!

Your drawing should look like this: this is the only line that you can make go through that point and be parallel to that line

Here’s the big question: Is that true in a spherical world like earth?

So basically we need to know: What is a line? Does it look like this?

Or does it take on the form of a projectile circling the globe Or does it take on the form of a projectile circling the globe? (like the equator?)

Well, some of the other ancient mathematicians decided to define a spherical line so that it is similar to the equator. This is called a great circle.

Great Circle: The intersection of a sphere and a plane that contains the center of the circle.

In spherical geometry, the equivalent of a line is called a great circle. Draw a line on your sphere then Make a conjecture about lines in spherical geometry. Euclidean Spherical Two points make a line. B B A A

If we draw another line on our sphere. Spherical What happened here that wouldn’t happen in Euclidean geometry? B A Look at the number of intersection points. Look at the number of angles formed. 2 8

Examples of great circles are the lines of longitude and the equator. In spherical geometry, then, a line is not straight - it is a great circle. Examples of great circles are the lines of longitude and the equator.

The equator is the only line of latitude that is a great circle. Lines of latitude do not work because they do not necessarily have the same diameter as the earth. The equator is the only line of latitude that is a great circle.

So what these guys figured out is that this geometry isn’t like Euclid’s at all. For instance - what about Parallel lines and his postulate? (we mentioned this earlier!)

Are lines of longitude or the equator parallel? NO! Are there any other great circles that are parallel? NO! So, what can you conclude from this? There are no parallel lines on a sphere!

What about perpendicular lines? Do we still have these? YES! The equator & lines of longitude form right angles! How many right angles are formed when perpendicular lines intersect? Draw two lines intersecting to form angles on a sphere. Draw two lines intersecting to form right angles on a sphere. 8! Four on the front side & four on the back.

What about triangles are there still triangles on a sphere?

B C A Draw a 3rd line on your sphere. Is this true in spherical geometry? In Euclidean Geometry, 3 lines usually make a triangle B B A C A C

But notice something about the spherical triangle But notice something about the spherical triangle. How many right triangles can it have? Is this true in Euclidian Geometry? B C A A C

We will stop here!!!!!

Estimate the 3 angles of your triangle. What about the angles of a triangle? Now move A and C to the equator. Move B to the top, what happens? Estimate the 3 angles of your triangle. The sum of the angles in a triangle on a sphere doesn’t have to be 180°! Let’s look at an example of this. Euclidean Spherical B B Find the sum of these angles. Make a conjecture about the sum of the angles of a triangle in spherical geometry. A A C C

What would happen if you moved A & C to opposite points on the great circle? What is the measure of angle B? What is the sum of the angles in this triangle? Could you get a larger sum? B 180º A C A C 360º

Think about this riddle: A bear leaves home - after going in a straight line for a while, he makes a 90 degree turn to the right. After another little while, he makes another 90 degree turn to the right and heads straight home. What color is the bear?

He’s white! He must be a polar bear who lives at the north pole and his little forage into the arctic has designed a triangle that has more than 180 degrees in its angles.

How does this effect our flight paths? What if you are traveling from a. LA to New York? Miami to Manila in the Philippines? Let’s look & see!