Navigation

Shortest paths on the ellipsoid.

• Destination from a start, course, and distance (geodesic)

  Where you end up after traveling a distance on an initial course along the ellipsoid, any length, with Karney's algorithm.

• Haversine distance (spherical)

  The great-circle distance on a sphere by the haversine formula, with its error against the ellipsoidal geodesic shown.

• Point part way along a great circle

  The point a given fraction of the way between two points along the great circle on a sphere, beside the same fraction along the ellipsoidal geodesic.

• Where two geodesic segments cross

  The crossing of the geodesics through two segments on the ellipsoid, nearest the segments' middles, and whether it falls within both segments or out on their extensions.

• Distance between two points (geodesic)

  The shortest distance and the start and end courses between two points on the WGS 84 ellipsoid (or any ellipsoid), exact to nanometers with Karney's algorithm.

• Midpoint on the geodesic

  The point halfway along the shortest path between two points on the ellipsoid, and the course there.

• Great-circle destination (spherical)

  Where a great circle from a start point on a course takes you after a distance on a sphere, and how far that is from the ellipsoidal answer.

• Great-circle distance and course (spherical)

  Distance and initial and final courses along the great circle on a sphere, with the difference from the ellipsoidal geodesic shown.

• Highest point of a geodesic (vertex)

  The vertex of a geodesic: the northernmost point it reaches, where it runs due east or west, and how far along the line from the start it lies.

• Vincenty direct (legacy)

  The destination by Vincenty's 1975 method, for checking legacy software; reports how far it lands from the Karney destination.

• Vincenty inverse (legacy)

  Distance and courses by Vincenty's 1975 method, for checking legacy software; reports its difference from Karney and fails honestly near antipodes.

• Waypoints along a route line

  Points along the geodesic (or rhumb line) from A to B at N equal intervals, a fixed spacing, or given fractions, with distance and course at each, as a table, a GPX route, and GeoJSON.

Horizon, line of sight, and radio clearance.

• Dip of the horizon

  How far below eye level the visible horizon lies from a height, with refraction, and the navigator's 1.76′√h rule with its difference.

• Fresnel zone and radio link clearance

  The first Fresnel zone radius at a point on a radio link, the 60% clearance it needs, and the Earth's bulge there, summed into the clearance above a smooth Earth.

• Distance to the horizon

  How far you can see from a height: the horizon distance with no refraction, with optical refraction, and for radio (4/3 Earth), with the rules of thumb and their errors.

• Can two points see each other?

  The farthest two raised points can see each other over the Earth's curve, and for a given distance whether they can, the clearance at the midpoint, and how much of the far target is hidden.

Constant-course lines.

• Destination on a constant course (rhumb line)

  Where a constant course (rhumb line, loxodrome) for a given distance ends on the ellipsoid; a rhumb that would cross a pole stops there.

• Rhumb line distance and course

  The constant course and distance of the rhumb line (loxodrome) between two points on the ellipsoid, and how much longer it is than the shortest route.

Routes, legs, and the geometry around them.

• Closest point on a route

  The point on a multi-leg route closest to a position: which leg, how far along the route, and how far off it (right of course positive).

• Where two courses cross

  The point where two courses from two positions meet, on geodesics or rhumb lines, and how far each has to run to get there.

• Closest point of approach

  When two moving objects come closest, how close, and the bearing and range then, in a local flat frame (for separations under 500 km), with climb rates and height for aircraft in 3D.

• Cross-track and along-track distance

  How far a point is off the course line from A to B (right of course positive), how far along the course its closest point lies, and that point, on the ellipsoid to the millimeter.

• Fly-by turn anticipation

  How early to start a fly-by turn at a waypoint: turn radius, lead distance, arc length, and time in the turn, from the inbound and outbound courses, speed, and bank angle or turn rate.

• Intercept a moving target

  The course to steer, the time, and the meeting point for reaching a target that is moving on a steady course and speed, or why it cannot be reached.

• Route legs, courses, and totals

  Each leg's distance, true and magnetic course, and cumulative distance for a route of waypoints, with leg times and arrival times from a groundspeed and departure time.

• Range rings

  Geodesic circles at one or more distances around a point, drawn true on the ellipsoid, with their areas, and a GeoJSON file cut correctly at the antimeridian and around a pole.

• Time, speed, and distance

  Solves for time, speed, or distance from the other two, and the arrival time from a departure time and UTC offset.

Distances and angles between points with heights.

• 3D distance between two points

  The straight-line distance between two points with heights, such as a drone and its ground station, through Earth-centered coordinates, with the ground distance, height difference, and elevation angle.

• Look angles to a target

  Azimuth, elevation, and straight-line range from an observer to a target, both with heights above the ellipsoid, with a warning when the Earth's curve hides the target.

• Vector sum, dot, and cross product

  Adds any number of 2D or 3D vectors head to tail; for two vectors, also their difference, dot and cross products, the angle between them, and the projection of one on the other.

• Vector components and direction

  Turns a magnitude and direction (and elevation, for 3D) into x, y, and z components, or components back into magnitude and direction, with the direction convention stated: navigational from north or mathematical from +x.