This handout contains all the math you need to do these tasks:

In all diagrams and equations:

Geometry of the flat Earth

Both distance and projection involve

The bearing is the angle made by a line drawn from A to B with another line drawn from A to the North Pole. Here’s a picture:

A step on the Earth’s surface from A to B

A step on the Earth’s surface from A to B

The homework has three computational problems related to this picture:

The first two problems can be solved using the following equations:

In case Δ y is zero, you want to use the two-argument arc-tangent function. In Beginning Student Language this is the atan function with two arguments. In the figure Δ x is about 60mm and Δ y is about -36mm and the bearing β is (atan 60 -36), which is about 120 degrees.

The third problem requires that you solve for θB and ϕB; given

I expect you to be able to solve for θB given R and Δ y, and similarly solve for ϕB given R and Δ x.

It is easy to get equations wrong. The only way you can know for sure is to test with actual locations. As a source of ideas you can make up your own coordinates, measure coordinates in the field, look up coordinates, or use Google Earth or Google Maps.

You can also use the Great Circle calculator at, but because the Great Circle calculator uses a more accurate model of the Earth, your answers will be off by a few percent.