geometrydistancepointwgs84great-circle

Point to great-circle segment distance

I want to calculate the distance from a point given by latitude and longitude to a line-segment given by two points (part of a great-circle). All coordinates are given in WGS84.

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I know how to calculate this in Cartesian coordinates but not on a sphere. Can somebody please provide the formula?

Solution

  • This is cross-track distance described here

    dxt = asin( sin(δ13) ⋅ sin(θ13−θ12) ) ⋅ R
    where
 δ13 is (angular) distance from start point to third point
     θ13 is (initial) bearing from start point to third point
     θ12 is (initial) bearing from start point to end point
     R is the earth’s radius

    You can calculate needed distance and bearings using formulas from given page

    distance
a = sin²(Δφ/2) + cos φ1 ⋅ cos φ2 ⋅ sin²(Δλ/2)
c = 2 ⋅ atan2( √a, √(1−a) )
d = R ⋅ c
where   
 φ is latitude, λ is longitude, R is earth’s radius (mean radius = 6,371km);

bearing
θ = atan2( sin Δλ ⋅ cos φ2 , cos φ1 ⋅ sin φ2 − sin φ1 ⋅ cos φ2 ⋅ cos Δλ )
    where
φ1,λ1 is the start point, φ2,λ2 the end point (Δλ is the difference in longitude)

    note that angles need to be in radians to pass to trig functions