1. Spherical Geometry

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

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

4. 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.

5. 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

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

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

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

9. 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.

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

11. 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

12. 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.

13. 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.

14. 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!)

15. •Are lines of longitude or the
equator parallel?
NO!
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?

16. •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?

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

18. 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

19. 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.

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