Rhumb line distance calculator

What a rhumb line is and what the response contains

A rhumb line, also called a loxodrome, is a curve on the sphere that crosses all meridians at a constant angle. On a Mercator chart it draws as a straight line, which made it the practical sailing course for centuries even though it is longer than the great-circle (orthodrome) path between the same two points except along the equator or a meridian. This calculator returns that surface length for a spherical Earth of radius R, together with the constant bearing from true north measured clockwise in degrees from zero to three hundred sixty. The numeric payload includes the normalised start and end latitudes and longitudes, distance in the requested unit, the same distance always echoed in metres as distance_m for unambiguous comparison, the unit tag, bearing_deg, the radius used in metres, the method tag rhumb_line, the decimal precision applied, and a short formula string so API consumers can document the model without reopening the source. There is no geodesic ellipsoid path here and no iterative Vincenty solver: the engine either evaluates the closed spherical rhumb formulas inside the supported coordinate ranges or rejects the input with invalid_input. Rejected cases include missing coordinates, non-finite values, latitude outside the open interval from negative ninety to ninety degrees, longitude outside the closed interval from negative one hundred eighty to one hundred eighty degrees, and exact poles where the isometric latitude becomes singular.

How coordinates, units, and radius are read so results stay reproducible

Latitude must lie strictly inside negative ninety to ninety degrees because the isometric latitude uses the tangent of forty-five degrees plus half the geodetic latitude, which is undefined at the poles; longitude must lie between negative one hundred eighty and one hundred eighty inclusive. 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 course length. An optional radius field sets the sphere radius in metres when you need a textbook Earth radius other than the default six million three hundred seventy-one thousand and eight point eight metres. Internally the engine converts degrees to radians, wraps the longitude difference into the principal interval from negative pi to pi, computes the isometric latitude difference as the natural log of the ratio of the two tan of pi-over-four-plus-half-latitude terms, forms q as delta-phi over delta-psi or cos of the start latitude when the track is essentially east-west, multiplies R by the square root of delta-phi squared plus q times delta-lambda squared, and obtains bearing from atan2 of delta-lambda and delta-psi. The preferred fields are lat1 and lon1 for the start point and lat2 and lon2 for the end point, all in decimal degrees, but aliases such as start_lat, end_lon, latitude1, and longitude2 are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. An optional unit field chooses metres, kilometres (the default), statute miles, or nautical miles, with longer aliases such as kilometers and nautical_miles normalised to short tags. Optional precision, with aliases decimals and dp, chooses how many decimal places to expose for distances and bearing, defaulting to six and capped at fifteen so JSON goldens stay stable. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing loxodrome formulas yourself.

Where constant-bearing distance fits in navigation study, charts, 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. Pair it with coordinate conversion or geohash tools when you need related geo steps, and with other parse endpoints when you are building a small lab of free, deterministic navigation helpers.