Great-circle path intersection

Aviation, sailing, and GIS work often describe a track as “from this fix, heading such-and-such degrees,” not as a pair of endpoints; the engine converts each point-plus-bearing into a great-circle plane on the unit sphere, intersects the planes, and returns both antipodal intersection coordinates.

What two point-and-bearing great circles mean and what the response contains

A great circle is the shortest path between points on a sphere and the track an aircraft or ship follows when flying a pure great-circle course between fixes. When you only know a single fix and the initial true bearing, that pair still defines the entire great circle: the local north and east unit vectors at the fix, combined with cosine and sine of the bearing, give a tangent direction, and the plane through the origin, the fix, and that tangent is unique. Two such planes generally meet in a line through the Earth’s center; that line pierces the sphere at two antipodal points, which are the only places both courses can share. This calculator returns the normalized inputs, the primary intersection latitude and longitude chosen as the candidate closer to the first origin (then the second origin on a distance tie), the antipodal latitude and longitude, the precision used, a method tag great_circle_plane_intersection, and a short formula reminder. There is no iterative ray marching and no chart projection: the engine either builds finite unit vectors and a non-zero cross product of the two plane normals or rejects the request with invalid_input. Rejected cases include missing coordinates or bearings, non-finite values, latitudes outside minus ninety to ninety, longitudes outside minus one hundred eighty to one hundred eighty, and parallel paths whose point-plus-bearing pairs define the same great-circle plane so that no unique intersection exists.

How latitudes, longitudes, and bearings are read and kept reproducible

Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces, hex prefixes, and non-numeric tokens are rejected with a clear message so a bad cell does not become a wrong fix. Bearings may be any finite real and are normalized into the half-open interval from zero inclusive to three hundred sixty exclusive, so negative headings and values above a full circle map cleanly without changing geometry. The preferred fields are lat1 and lon1 with bearing1 for the first path and lat2 and lon2 with bearing2 for the second, but aliases such as start1_lat, latitude1, phi1, brng1, course1, heading1, and the matching second-path names are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose for latitudes, longitudes, and bearings, defaulting to six and capped at fifteen so JSON goldens stay stable. Computation converts each origin to an Earth-centered unit vector, builds the local north and east basis, forms the tangent direction for each bearing, takes plane normals as the cross product of position and direction, and intersects those normals; if the magnitude of the resulting intersection vector is at or below a tiny epsilon, the planes are parallel and the request fails.

Where path intersection fits in navigation, GIS, and software

On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result.