Parallel+Lines

Parallel Lines
As we learned in the section on __Straight Lines__, a line in spherical geometry is defined as a great circle, which contains two points on a sphere and the center of the sphere. We conclude that each line slices the sphere in exact halves. So, if two lines (great circles) were drawn on a sphere they will always intersect each other on the opposite side. Therefore, parallel lines on a sphere do not exist. See figures below for examples of intersecting great circles. The reason that there are no parallel lines is that lines (i.e. great circles) on a sphere all contain the center of the sphere so the planes that create the great circles MUST intersect because they have a point in common (i.e. the center of the sphere). Because those planes intersect, they can not be parallel so there are no parallel lines on a sphere.



The figures allow for a more visual view of why parallel lines on a sphere do not exist.

Materials needed: spheres (balls, globe, etc.), rubber bands (also, string, wire, etc. would work for bigger spheres) Wrap one rubber band positioned at a spot of your choice around the ball. If the rubber band stays on the ball then it must be in the middle of the ball, forming a great circle. Then attempt to place a second rubber band around the ball so that it also forms a great circle. Can you position both rubber bands where they do not intersect, forming parallel lines?”
 * Activity 1:**


 * Claim:** Lines cross each other either two times.


 * Proof:** Let L be a line and a point A which is not on the given line L. All lines through A will pass through L. Therefore, lines cross each other two times. Q.E.D




 * Activity 2:**

Using the applet, [], create lines (great circles) on the spheres and adjust, rotate, move, etc. these lines to see that lines cross each other two times. Also, play with the intersection tools that are on the applet as well. Here is an example below: