# Calculation Methods

The Geographic Midpoint Calculator provides you three methods for calculating the midpoint for two or more points on the earth's surface. Each of the three methods has its own advantages, and the one that you choose depends on what you want to use it for.

## A. Geographic midpoint

The geographic midpoint, sometimes called the center of gravity, center of mass or centroid is the average coordinate for a set of points on a spherical earth. If you mark several points on a world globe along with the geographic midpoint, the geographic midpoint does indeed look like it is in the center. Imagine that several weights are placed at various points on a world globe and then the globe is allowed to rotate freely until the heaviest part of the globe is pulled by gravity until it is facing downward. Then the lowest point on the globe would be the geographic midpoint for all of the weighted locations.

## B. Center of minimum distance

This method finds the location that minimizes the combined travel distance from a set of points on a spherical earth. The center of minimum distance is the point of absolute minimum travel distance, however it does not attempt to equalize the amount of distance traveled from the other points.

## C. Average latitude/longitude

This is simply the mathematical average of the latitudes and longitudes of a set of points. This is equivalent to finding the midpoint on a flat rectangular projection map. When the distance between the points is less than 250 miles (400 km) this method gives a close approximation to the geographic midpoint in Method A.

## Advantages and disadvantages

Sometimes the geographic midpoint and the center of minimum distance will be fairly close together, it depends on the geometric layout of the points. For example, the geographic midpoint and the center of minimum distance for Washington DC, Detroit and Nashville are only 57 miles (92 km) apart. Often, however there is a considerable separation between the two midpoints.

Interestingly, the center of minimum distance for Seattle, Minneapolis and St. Louis is located in Minneapolis. In other words if three people, one in each of these cities want to get together, the shortest overall travel distance is achieved if the person in Minneapolis stays home, and the person in Seattle and the person in St. Louis travel to Minneapolis. When there are less than 10 places, the center of minimum distance is often located in or near one of the existing places. When there are more than 10 places, the center of minimum distance tends to be located somewhere between places.

In contrast, the geographic midpoint from Method A for Seattle, Minneapolis and St. Louis is located in South Dakota. If you look at a map, this midpoint does appear to be in the middle of the three cities.

So how do you decide which midpoint to choose? Let's suppose that the three people mentioned above are employees of the same company, and that they will meet together in a conference with the company paying all expenses. Generally a business or a single individual will want to minimize overall expenses, and the center of minimum distance method is well suited to accomplish this. In this case the lowest cost alternative is to have the conference in Minneapolis. Note that the calculation is for air distance; in the case of these three cities the air distance also correlates fairly well with highway distance.

Now let's suppose that there are three families, one in Seattle, one in Minneapolis and one in St. Louis, and that they want to get together for a family reunion. Since each family is paying its own expenses, now the fairness issue comes into play. It would probably be more fair to have the family reunion in South Dakota, even though the overall travel distance is somewhat higher (about 87 miles or 140 km per family in this case). In general for three or more points, the geographic midpoint method seeks to find the point of balance, and equalizes the distances better than the center of minimum distance method, although in real life there are rarely situations where distances are going to be exactly equal. You may want to calculate both Methods A and B for comparison purposes.

Methods A and B are used most of the time because they are modeled on a spherical earth and give the best indication of travel distance. If you have read the Phoenix, Arizona-Kabul, Afghanistan example on the home page you know that a true midpoint can end up in the polar regions, especially if your locations are on different continents. If you run into this situation you can try Method C Average latitude/longitude. It doesn't find minimum or equal travel distances, but it helps assure that the midpoint is at the same latitude as your other points and might be a pleasant alternative if Methods A and B are telling you that your next convention should be held in the middle of Greenland.

## Geographic Midpoint Calculator

See how the midpoint calculations can work in real life with this free online application. With the Geographic Midpoint Calculator you can find the midpoint of an airline flight between two cities. You can also find your personal center of gravity by selecting all of the locations that you have lived in, then view all of those locations displayed on a map along with a marker pointing at the location of your exact personal center of gravity.

## Let's meet in the middle

This free tool finds the ideal restaurant or other point of interest halfway between two or more addresses. Meet your friend or business acquaintance for lunch.