Concept Byte: Exploring Spherical Geometry

Use With Lesson 3-5

ACTIVITY

Euclid was a Greek mathematician who identified many of the definitions, postulates, and theorems of high school geometry. Euclidean geometry is the geometry of flat planes, straight lines, and points.

In spherical geometry, the curved surface of a sphere is studied. A “line” is a great circle. A great circle is the intersection of a sphere and a plane that contains the center of the sphere.

Activity 1

You can use latitude and longitude to identify positions on Earth. Look at the latitude and longitude markings on the globe.

1. Think about “slicing” the globe with a plane at each latitude. Do any of your “slices” contain the center of the globe?
2. Think about “slicing” the globe with a plane at each longitude. Do any of your “slices” contain the center of the globe?
3. Which latitudes, if any, suggest great circles? Which longitudes, if any, suggest great circles? Explain.

You learned in Lesson 3.5 that through any point not on a line, there is one and only one line parallel to the given line (Parallel Postulate). That statement is not true in spherical geometry. In spherical geometry,

through a point not on a line, there is no line parallel to the given line.

Since lines are great circles in spherical geometry, two lines always intersect. In fact, any two lines on a sphere intersect at two points, as shown below.

Table of Contents