Destination point calculator

Worked example: northeast from Denver

Start on the Colorado state capitol and head 45°, due northeast. The table shows where a great circle and a rhumb line end for four distances, and how far apart those two ends are.

At 100 miles the two methods agree to within about a mile, which is why short trips can ignore the difference. By 1,000 miles they are tens of miles apart, and by 3,000 miles the great circle has swung far to the east, arriving on a heading close to due east while the rhumb line is still holding 45° and has climbed much farther north.

How to use it

1. Pick the Start point by search, coordinates, Use GPS, or a click on the map.
2. Type the Bearing in degrees from true north, such as 45, or as a compass point such as NE or WSW.
3. Enter the Distance and choose the unit: mi, km, or nm.
4. Read the great-circle end point, the arrival heading, and the rhumb-line end in the readout. The map draws both paths. Press Copy destination coordinates to copy the result, or copy the page link to share it.

The formula

The great-circle destination is the direct problem of spherical trigonometry. With start latitude φ1, longitude λ1, bearing θ, and angular distance δ (the distance divided by the Earth's radius, 6,371.0088 km):

φ2 = asin(sin φ1 · cos δ + cos φ1 · sin δ · cos θ)
λ2 = λ1 + atan2(sin θ · sin δ · cos φ1, cos δ - sin φ1 · sin φ2)

The final bearing is the direction of travel when you arrive. On a great circle it changes along the way, except when you travel along a meridian or the equator. Heading northeast from the northern hemisphere, the path bends toward the east as it goes, so the arrival heading is larger than the starting bearing.

Great circle or rhumb line

A great circle is the shortest path over the sphere, but following it means changing heading all the time. A rhumb line keeps one compass heading the whole way. It crosses every meridian at the same angle and draws as a straight line on a Mercator map, which is why sailors used it for centuries. For the same start, bearing, and distance the two end in different places, and the readout shows the gap.

A rhumb line on any heading other than due east or west spirals toward a pole. If the distance is long enough to reach the pole first, the tool says so instead of inventing an end point.

Accuracy and limits

The Earth is slightly flattened, and a sphere model can be off by up to about 0.5 percent of the distance compared with the WGS84 ellipsoid. For a 100 mile leg that is under half a mile, fine for planning but not for surveying. Bearings are relative to true north. A magnetic compass differs by the local declination, which is more than 10 degrees in parts of the US, so correct for it before you follow a bearing in the field.

Distances are capped at 20,000 km, just under half the Earth's circumference. Beyond that the shorter way to the end point runs the other way round the globe, and the final bearing would no longer describe the path you set out on.