Latitude Longitude Midpoint Calculator
Calculate the geographic midpoint between two coordinates on Earth using spherical trigonometry. Accounts for Earth curvature for accurate results.
Latitude Longitude Midpoint Calculator
Calculator
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Formula: midLat = atan2(sin(lat1)+sin(lat2), sqrt((cos(lat1)+Bx)^2 + By^2))
Worked example — Midpoint: approximately 52.33 N, 36.84 W (over the Atlantic Ocean, south of Greenland)
Formula
midLat = atan2(sin(lat1)+sin(lat2), sqrt((cos(lat1)+Bx)^2 + By^2))
The spherical midpoint formula uses Bx = cos(lat2)*cos(dLon) and By = cos(lat2)*sin(dLon) as intermediate values. The midpoint latitude is found using atan2 for proper quadrant handling. The midpoint longitude is computed by adding atan2(By, cos(lat1)+Bx) to the first longitude. This method correctly handles all cases including crossing the antimeridian.
Worked Examples
Example 1: Midpoint Between New York and London
Problem:Find the geographic midpoint between New York (40.7128 N, 74.006 W) and London (51.5074 N, 0.1278 W).
Solution:Convert to radians, compute Bx = cos(lat2)*cos(dLon), By = cos(lat2)*sin(dLon). midLat = atan2(sin(lat1)+sin(lat2), sqrt((cos(lat1)+Bx)^2 + By^2)) midLon = lon1 + atan2(By, cos(lat1)+Bx)
Result:Midpoint: approximately 52.33 N, 36.84 W (over the Atlantic Ocean, south of Greenland)
Example 2: Midpoint Between Los Angeles and Tokyo
Problem:Calculate the midpoint between LA (33.9425, -118.4081) and Tokyo (35.6762, 139.6503).
Solution:Apply spherical midpoint formula. Total distance is approximately 8,815 km. Midpoint lies along the great circle route over the Pacific.
Result:Midpoint: approximately 49.02 N, 168.85 W (over the North Pacific)
Frequently Asked Questions
What is the geographic midpoint between two coordinates?
The geographic midpoint is the point on Earth's surface that lies exactly halfway along the great circle path between two locations. Unlike a simple arithmetic average of coordinates, the geographic midpoint accounts for the curvature of the Earth using spherical trigonometry. This means the midpoint of two locations on the same latitude will not necessarily have the same latitude, especially for points far apart. The calculation converts coordinates to 3D Cartesian vectors, averages them, and converts back to geographic coordinates.
Why is the midpoint not just the average of two coordinates?
Simply averaging latitude and longitude values produces incorrect results because the Earth is a sphere, not a flat surface. Longitude lines converge at the poles, so one degree of longitude represents a much shorter distance near the poles than at the equator. Additionally, when two points are on opposite sides of the 180-degree meridian, a simple average would place the midpoint on the wrong side of the globe. The spherical midpoint formula handles these edge cases correctly by working in 3D Cartesian space.
What are practical uses for finding the geographic midpoint?
The geographic midpoint has many practical applications. It is commonly used to find a fair meeting place between two people in different cities, to plan refueling stops for long-distance flights, and to determine the center point of a delivery service area. Scientists use midpoint calculations when studying migration patterns, tracking storms, and analyzing satellite coverage areas. In logistics, finding the midpoint helps optimize warehouse placement between suppliers and customers.
Does the midpoint calculation work for antipodal points?
Antipodal points are locations on exact opposite sides of the Earth, such as the North and South poles. For perfectly antipodal points, the midpoint is mathematically undefined because every great circle connecting them has the same length, so there are infinite midpoints forming a complete circle around the Earth. In practice, most real-world coordinate pairs are not perfectly antipodal, and the formula will return a valid midpoint. If your inputs are very close to antipodal, the result may be numerically unstable.
How does the vector method compute a great-circle midpoint?
Each coordinate is converted to a unit vector with x equal to cos(latitude) times cos(longitude), y equal to cos(latitude) times sin(longitude), and z equal to sin(latitude). The two vectors are summed and the result normalised back to unit length, which places it on the sphere halfway along the arc. Converting back uses atan2 for longitude and the arcsine of z for latitude; atan2 is what keeps the quadrant correct.
What is the difference between a midpoint of two places and the centroid of several?
The two-point midpoint lies exactly on the great circle joining them, equidistant from both. A centroid of three or more points is the normalised average of all their unit vectors and is generally not equidistant from any of them; it is pulled toward clusters. For meeting-point problems with several parties, the centroid can be a poor answer, and a point minimising total or maximum travel distance is usually more useful.
Why does the initial bearing change as you travel along a great circle?
A great circle crosses successive meridians at different angles, so the compass heading shifts continuously along the route even though the path is the shortest one on the sphere. The bearing at the origin is the initial bearing; the bearing on arrival differs, sometimes by a great deal on long high-latitude routes. This is why great-circle flying is flown as a series of legs, each with a recalculated heading.
How much does a spherical midpoint differ from a true ellipsoidal one?
Earth is an oblate ellipsoid, flattened by about one part in 298 under the WGS84 definition, so a perfect-sphere calculation carries a small systematic error. For midpoints the discrepancy is typically well under a kilometre and is largest for long paths at high latitudes. Spherical results are fine for meeting points, mapping and general navigation, while surveying and geodetic work should use a geodesic solution on the ellipsoid.
How should longitudes be handled when the two points straddle the antimeridian?
Enter longitudes normalised to the range minus 180 to plus 180 degrees and let the vector method do the rest. Because the coordinates are converted into three-dimensional components before averaging, the 180 degree discontinuity disappears entirely and the atan2 conversion returns a correctly wrapped longitude. Naive averaging of, say, 175 degrees east and 175 degrees west would give zero degrees, placing the midpoint on the wrong side of the planet.
When is a rhumb line halfway point more useful than the great-circle midpoint?
A rhumb line, or loxodrome, holds a constant compass bearing and is longer than the great circle, but it is far easier to steer. Its halfway point is what you want when planning a route that will actually be sailed or flown on a fixed heading, or when working on a Mercator chart where rhumb lines are straight. Use the great-circle midpoint when the question is about shortest distance or a fair meeting location.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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