Ideas in Geometry/Spherical Geometry

Basic Points

It is important to recognize and understand these key concepts to fully expand upon properties of spherical geometry.

A point: a point on a sphere is the same as a point on a plane.
A line: a line on a sphere is called an arc due to the shape of a sphere. It is also the shortest distance between two points on the sphere . If an arc is extended, it will form a great circle. A great circle, however is the end of the lines path.
A great circle: a great circle refers to arcs that are extended to form a circle between two points. Because the arcs are the shortest distance between the two points, the circle being formed will revolve around the entire sphere, as if its radius where the very center inside of the sphere. This circle always slices a sphere in half. There are infinitely many great circles on a sphere.
Antipodal points: these points are exactly opposite of each other on the sphere, such as the poles. All great circles run through antipodal points.
A circle: Every point has to be the same distance from the center, referencing its radius.

Parallel Lines

In spherical geometry Parallel lines DO NOT EXIST.

In Euclidean geometry a postulate exists stating that through a point, there exists only 1 parallel to a given line.

However, it is important that we remember the definition of a line in spherical geometry. Because a line is defined as a great circle, we can understand that each line goes through two antipodal points, slicing the sphere in half (shown in the first picture). If two lines (great circles) were to be drawn, they will always cross paths as they cross over to the opposite hemisphere (shown through lines A and B in picture 2).

Circle C appears to be parallel to line A. However, Circle C cannot be considered a line, since a line must be an arc that can be extended to run through two antipodal points. Since Circle C does not, it is not a parallel line to A. Instead we can call circle C simply a line of latitude.

Therefore, Parallel lines do not exist since any great circle (line) through a point must intersect our original great circle.

Angles and Triangles

Angles and their properties exist the same in spherical geometry as they do in Euclidean Geometry. An angle in spherical geometry is simply formed by two great circles. Thus, in picture 2 up above, there are angles formed where lines A and B intersect.

A triangle however, is different. In Euclidean Geometry, the sum of the interior angles of a triangle must equal up to 180°, since lines on a plane are very constricted.
In spherical geometry, a triangle is formed by three arcs of great circles intersecting. These three arcs can form triangles with interior angle sums of much larger than 180 degrees. For example figure below shows three great circles. One is the line that lies along the equator, and the two other lines are perpendicular to the equator, but they both run through the same antipodal point at the top of the sphere. Therefore, we know that all three lines intersect to form a triangle and we also know that the two lines intersecting the equator already create an angle of 90° each. Thus, we can determine that the interior angle sum of the triangle is already larger than 180°.

In fact, the angle sum can be anywhere from 180° to 540°. We can see how the size of the angles increase and decrease if we examine smaller and larger triangles. The larger the triangle becomes on the sphere, the more Euclidean properties it loses. Thus, the angle sum will become larger.

Picture References

http://en.wikiversity.org/wiki/File:Intersecting_great_circles.png
http://commons.wikimedia.org/wiki/File:Linalg_great_circle.png
http://www.abc.net.au/science/morebigquestions/sphere.htm
http://euler.slu.edu/escher/index.php/File:Great-circles.png