Here are solutions to the direct and inverse geodesy problems using haversines formula on the sphere and Vincenty's formulae on the ellipsoid as models of the Earth.
{{ The inputs for the direct or forward problem in geodesy.
* {x} The departure point on the ellipsoid.
* {α₁} The azimuth from the departure point.
* {s} The distance to the arrival point. }}
unique type DirectProblem a α s = { x : a, α₁ : α, s : s }
{{ The outputs for the solution to the direct or forward problem in geodesy.
* {y} The arrival point.
* {α₂} The azimuth at the arrival point. }}
unique type DirectSolution a α = { y : a, α₂ : Optional α }
InverseProblem.doc =
{{ The inputs for the inverse or reverse problem in geodesy.
* {x} The departure point.
* {y} The arrival point. }}
unique type InverseProblem a = { x : a, y : a }
{{ The outputs for the solution to the inverse or reverse problem in geodesy.
* {s} The distance between departure and arrival points.
* {α₁} The azimuth at the departure point.
* {α₂} The azimuth at the arrival point. }}
unique type InverseSolution s α = InverseSolution s α (Optional α)
The Haversine transcript finds the distance from Big Ben, London to the Statue of Liberty, New York, with inputs in decimal degrees and outputs in metres.
The Vincenty transcript finds distances and bearings between points from Vincenty's 1975 paper "Direct and Inverse Solutions of Geodesics on the Ellipsoid with Applications of Nested Equations".