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A Follow-Up of Lesson 3-6

Non-Euclidean Geometry

So far in this text, we have studied

plane Euclidean geometry

, which is based on a

system of points, lines, and planes. In

spherical geometry

, we study a system of

points, great circles (lines), and spheres (planes). Spherical geometry is one type of

non-Euclidean geometry

.

Plane Euclidean Geometry

Spherical Geometry

Longitude lines

and the equator

A

model great

m

A great circle

circles on Earth.

divides a sphere

P

P

into equal halves.

P

E

Plane

contains line

and point A not on .

Sphere

E

contains great

Polar points are endpoints of a

circle m and point P not

diameter of a great circle.

E

on m. m is a line on sphere

.

The table below compares and contrasts lines in the system of plane Euclidean

geometry and lines (great circles) in spherical geometry.

Plane Euclidean Geometry

Spherical Geometry

Lines on the Plane

Great Circles (Lines) on the Sphere

1. A line segment is the shortest path

1. An arc of a great circle is the shortest

between two points.

path between two points.

2. There is a unique line passing through

2. There is a unique great circle passing

any two points.

through any pair of nonpolar points.

3. A line goes on infinitely in two directions.

3. A great circle is finite and returns to its

original starting point.

4. If three points are collinear, exactly one

4. If three points are collinear, any one of

is between the other two.

the three points is between the other two.

A is between B and C.

C

A

B

C

B is between A and C.

C is between A and B.

B is between A and C.

A

B

In spherical geometry, Euclid’s first four postulates and their related theorems

hold true. However, theorems that depend on the parallel postulate (Postulate 5)

may not be true.

In Euclidean geometry parallel lines lie in the same plane and never

intersect. In spherical geometry, the sphere is the plane, and a great circle

A

represents a line. Every great circle containing A intersects . Thus, there

exists no line through point A that is parallel to .

(continued on the next page)

Investigating Slope-Intercept Form 165

Geometry Activity Non-Euclidean Geometry 165

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