How to Find the Distance in Spherical Geometry

The Earth is an example of a spherical geometry.

Spherical geometry is a type of geometry that is based on the two-dimensional surface of a sphere. Spherical geometry can differ greatly from the straight line geometry found in a Euclidean geometry, as straightforward concepts such as straight lines can have very different definitions. For example, distance in spherical geometry is measured along what is called a great circle, which is a curved line that follows the radius of the sphere. Measuring distance using the great circle depends on converting the spherical coordinates to the more common Cartesian coordinates, and calculating the distance in this fashion.

Instructions

1. Determine the latitude and longitude of two separate points located on the sphere. As an example, assume the two points are (45, 180) and (30, 45), where the coordinates given are for the latitude and longitude, respectively, in degrees.

2. Determine the sine of the latitude of both coordinates and multiply the two numbers together. For example, the sine of 45 degrees is 0.707 and the sine of 30 degrees is 0.5 (these values can be determined using the "sine" function on any scientific calculator). Multiplying these two numbers together gives 0.3535. Call this result A.

3. Determine the cosine of the latitude of the two points and multiply the result together. For example, the cosine of 45 degrees is 0.707 and the cosine of 30 degrees is 0.866 (these values can be determined using the "cosine" function on any scientific calculator). The product of these two numbers is 0.6128. Call this result B.

4. Find the cosine of the difference between the longitude of the two points. The two longitude values are 180 degrees and 45 degrees, so taking the difference of the two values gives 135 degrees. The cosine of 135 degrees is -0.707. Call this result C.

5. Multiply result B and result C together. For example, multiplying 0.6128 by -0.707 gives -0.4332. Call this result D.

6. Add result A and result D together. The sum of -0.4332 and 0.3535 equals -0.079815. Call this result E.

7. Calculate the inverse cosine of result E, in radians, which is usually calculated on a scientific calculator using the "second function" of the "cosine" button. The inverse cosine of -0.079815 is 1.6507 radians. Call this result F.

8. Multiply result F by the radius of the sphere. If we use the Earth as an example, multiplying the radius of the Earth, which is about 6400 kilometers, by 1.6507 gives 10,564 kilometers. This is the distance between the two points in spherical geometry.

Tips & Warnings

• The equation given above requires that the latitude and longitude be given in units of degrees.

• The latitude (also called the polar angle) cannot exceed 90 degrees and the longitude (also called the azimuthal angle) cannot exceed 360 degrees.

References

• Wolfram MathWorld: Great Circle
• University of North Carolina Charlotte: Spherical Geometry