Spherical 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. 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? (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: For a given sphere, the intersection of the sphere and a plane that contains the center of the sphere.

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

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

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.

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

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

What about triangles are there still triangles on a sphere? Let’s look!

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

What about the angles of a triangle? Now move A and C to the equator. Move B to the top, what happens? Euclidean Spherical B C A A B C Estimate the 3 angles of your triangle. Find the sum of these angles. Make a conjecture about the sum of the angles of a triangle in spherical geometry. 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.

What would happen if you moved A & C to opposite points on the great circle? A B C AC What is the measure of angle B? What is the sum of the angles in this triangle? Could you get a larger sum? Triangle sum : 180º 360º Can be greater than 180º less than 540º

Line segment arc unique Great circle straight finite one Point = point; Line = Great Circle; Plane = sphere

Spherical Geometry

Spherical Geometry Lesson DFW-BKK (Bangkok) OPF-MNL (Miami-Philippines) LAX-MXP (LA – Milan) DFW-SIN (Singapore) LAX-JFK (LA-NY) LHR-SYD (London-Sydney)