A circle on the surface of the Earth which lies in a plane passing through the Earth's centre. Every two distinct and non-antipodal points on the surface of the Earth define a Great Circle.

It is internally represented as its normal vector - i.e. the normal vector to the plane containing the great circle.

see IsGreatCircle.

Class for data from which a GreatCircle can be computed.

position (n-vector) on the great circle

bearing from gcPos.

A closed segment of GreatCircle. It represent the shortest path on the surface of the Earth between the two positions.

see IsGreatArc.

Class for data from which a GreatArc can be computed.

start position (n-vector) of the great arc

end position (n-vector) of the great arc

alongTrackDistance p ga r how far position p is along a path described by great arc ga: if a perpendicular is drawn from p to the great arc, the along-track distance is the signed distance from the start point to where the perpendicular crosses the path.

    let p = decimalLatLong 53.2611 (-0.7972)
    let ga = greatArc (decimalLatLong 53.3206 (-1.7297)) (decimalLatLong 53.1887 0.1334)
    alongTrackDistance p ga r84 -- 62.3315757 kilometres

alongTrackDistance using the mean radius of the WGS84 reference ellipsoid.

angularDistance p1 p2 n computes the angle between the horizontal positions p1 and p2. If n is Nothing, the angle is always in [0..180], otherwise it is in [-180, +180], signed + if p1 is clockwise looking along n, - in opposite direction.

antipode p computes the antipodal horizontal position of p: the horizontal position on the surface of the Earth which is diametrically opposite to p.

crossTrackDistance p gc r computes the signed distance from horizontal position p to great circle gc. Returns a negative Length if position if left of great circle, positive Length if position if right of great circle; the orientation of the great circle is therefore important:

    let gc1 = greatCircle (decimalLatLong 51 0) (decimalLatLong 52 1)
    let gc2 = greatCircle (decimalLatLong 52 1) (decimalLatLong 51 0)
    crossTrackDistance p gc1 r84 = (- crossTrackDistance p gc2 r84)

    let p = decimalLatLong 53.2611 (-0.7972)
    let gc = greatCircleBearing (decimalLatLong 53.3206 (-1.7297)) (decimalDegrees 96.0)
    crossTrackDistance p gc r84 -- -305.663 metres

crossTrackDistance using the mean radius of the WGS84 reference ellipsoid.

destination p b d r computes the destination position from position p having travelled the distance d on the initial bearing (compass angle) b (bearing will normally vary before destination is reached) and using the earth radius r.

    let p0 = ecefToNVector (ecefMetres 3812864.094 (-115142.863) 5121515.161) s84
    let p1 = ecefMetres 3826406.4710518294 8900.536398998282 5112694.233184049
    let p = destination p0 (decimalDegrees 96.0217) (metres 124800) r84
    nvectorToEcef p s84 = p1

destination using the mean radius of the WGS84 reference ellipsoid.

finalBearing p1 p2 computes the final bearing arriving at p2 from p1 in compass angle.

Compass angles are clockwise angles from true north: 0 = north, 90 = east, 180 = south, 270 = west.

The final bearing will differ from the initialBearing by varying degrees according to distance and latitude.

Returns Nothing if both horizontal positions are equals.

initialBearing p1 p2 computes the initial bearing from p1 to p2 in compass angle.

Compass angles are clockwise angles from true north: 0 = north, 90 = east, 180 = south, 270 = west.

Returns Nothing if both horizontal positions are equals.

interpolate p0 p1 f# computes the horizontal position at fraction f between the p0 and p1@.

Special conditions:

    interpolate p0 p1 0.0 = p0
    interpolate p0 p1 1.0 = p1

errors if f || f 1

    let p1 = latLongHeight (readLatLong "53°28'46''N 2°14'43''W") (metres 10000)
    let p2 = latLongHeight (readLatLong "55°36'21''N 13°02'09''E") (metres 20000)
    interpolate p1 p2 0.5 = decimalLatLongHeight 54.7835574 5.1949856 (metres 15000)

Computes the intersection between the two given GreatArcs.

see also intersections

    let spd = kilometresPerHour 1000
    let t1 = Track (decimalLatLong 51.885 0.235) (decimalDegrees 108.63) spd
    let t2 = Track (decimalLatLong 49.008 2.549) (decimalDegrees 32.72) spd
    let oneHour = hours 1
    let ga1 = greatArc (t1, oneHour)
    let ga2 = greatArc (t2, oneHour)
    intersection ga1 ga2 = Just (decimalLatLong 50.9017225 4.494278333333333)

Computes the intersections between the two given GreatCircles. Two GreatCircles intersect exactly twice unless there are equal (regardless of orientation), in which case Nothing is returned.

    let gc1 = greatCircleBearing (decimalLatLong 51.885 0.235) (decimalDegrees 108.63)
    let gc2 = greatCircleBearing (decimalLatLong 49.008 2.549) (decimalDegrees 32.72)
    let (i1, i2) = fromJust (intersections gc1 gc2)
    i1 = decimalLatLong 50.9017226 4.4942782
    i2 = antipode i1

isBetween p ga determines whether position p is between start and end points of great arc ga. If p is not on the great arc, returns whether p is within the area bound by perpendiculars to the great arc at each point (in the same hemisphere).

isInsideSurface p ps determines whether the p is inside the polygon defined by the list of positions ps. The polygon is closed if needed (i.e. if head ps /= last ps).

Uses the angle summation test: on a sphere, due to spherical excess, enclosed point angles will sum to less than 360°, and exterior point angles will be small but non-zero.

Always returns False if ps does not at least defines a triangle.

    let malmo = decimalLatLong 55.6050 13.0038
    let ystad = decimalLatLong 55.4295 13.82
    let lund = decimalLatLong 55.7047 13.1910
    let helsingborg = decimalLatLong 56.0465 12.6945
    let kristianstad = decimalLatLong 56.0294 14.1567
    let polygon = [malmo, ystad, kristianstad, helsingborg, lund]
    let hoor = decimalLatLong 55.9295 13.5297
    let hassleholm = decimalLatLong 56.1589 13.7668
    isInsideSurface hoor polygon = True
    isInsideSurface hassleholm polygon = False

mean ps computes the mean geographic horitzontal position of ps, if it is defined.

The geographic mean is not defined for antipodals position (since they cancel each other).

Special conditions:

    mean [] = Nothing
    mean [p] = Just p
    mean [p1, p2, p3] = Just circumcentre
    mean [p1, .., antipode p1] = Nothing

surfaceDistance p1 p2 computes the surface distance (length of geodesic) between the positions p1 and p2.

surfaceDistance using the mean radius of the WGS84 reference ellipsoid.