Straight+Lines

Straight Lines
We know from Euclidean geometry that the shortest distance between any two points is a **__straight line__**. Using this definition, we will determine the characteristics of a straight line in spherical geometry.

Consider two cities on a globe, San Francisco and Rome. To find the shortest distance (line) between these two points it appears that one would draw the line in pink from San Francisco to Rome. Notice that this line is also shown on the flat map below the globe. However, the shortest distance between these two cities is actually the line in yellow.
 * __Exercise:__**

Great circle distance (yellow line) = 6,375 miles Mercator straight line (pink line) = 6,990 miles



__**Activity 1:**__ Materials needed: Globe and a piece of string. Instructions: Use the string to create a straight line from San Francisco to Rome similar to the line in pink above. Next, while holding the ends of the string on San Francisco and Rome, slide the string up the globe to the position indicated by the yellow line above. You will notice that there is excess string as you move closer to the position of the yellow line. The yellow line represents the shortest distance between the two cities and is a straight line on the sphere. Repeat the activity with any two cities on the globe to determine the path of the shortest distance (i.e. line) between the cities. Notice that the line between San Francisco and Djibouti, Ethiopia travels nearly over the North Pole.

Note that a continuous straight line on a sphere returns to its origin unlike a straight line on a plane which will never return to the origin.

Straight lines on a sphere are called __**Great Circles**__. A great circle is the intersection of a plane containing two points on a sphere and the center of the sphere. There are an infinite number of great circles on a sphere but the most familiar examples are the equator and lines of longitude. Latitude lines, however, are not great circles. Although latitude lines connect two points on a sphere, they do NOT include the center of the sphere.

It is no accident then that flight paths occur along great circles because they are the shortest distance between any two cities.



__**﻿Activity 2:**__ ﻿Materials Needed: Globe and a piece of string. Instructions: Choose any two points (cities) on the globe. Determine whether or not these cities lie on a great circle. Are there any points on a globe that do not lie on a great circle? Can you find two points on a globe which lie on more than one great circle? What characteristics must points on a sphere have to lie on more than one great circle?

__**Discussion:**__ All points on a sphere lie on at least one great circle, and the great circle is the shortest distance between the points. Recall from Euclidean geometry that the shortest distance between any two points is a straight line. Therefore, a great circle is a "straight line" on a sphere. Points that are 180 degrees apart on a sphere lie on an infinite number of great circles. For example, the north and south poles are 180 degrees apart and lie on all the lines of longitude. Other points on a sphere that are 180 degrees apart also lie on an infinite number of great circles which means they lie on an infinite number of straight lines.