Distance from Lat/Lng point to Minor Arc segment

Question

I need to calculate the shortest distance from a lat/lng GPS point P to a line segment described by 2 other lat/lng GPS points A and B.

'Cross-track distance' helps me to calculate the shortest distance between P and the great circle described by A and B.

However, this is not what I want. I need need the distance between P and the line segment of A-B, not the entire great circle.

I have used the following implementation from http://www.movable-type.co.uk/scripts/latlong.html

Formula:    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

The following images hopefully demonstrate the problem I am trying to solve:

In the first image the Cross-Track distance, indicated by the green line is correct and indeed the shortest distance to the line segment AB.

In the second image the problem with cross-track distance is shown, In this case I would want the shortest distance to be the simple distance AP, but Cross-Track distance gives me the distance indicated by the red line.

How do I change my algoritm to take this into account, or check whether or not point X is within AB. Is it possible to do this computationally? Or is iterative the only possible (expensive) solution? (take N points along AB and calculate the min distance from P to all these points)

For simplicity purposes all lines in the images are straight. In reality, these are minor arcs on a great circle

Answer 1

First, some nomenclature:
Our arc is drawn from p1 to p2.
Our third point is p3.
The imaginary point that intersects the great circle is p4.
p1 is defined by lat1,lon1; p2 by lat2,lon2; etc.
dis12 is the distance from p1 to p2; etc.
bear12 is the bearing from p1 to p2; etc.
dxt is cross-track distance.
dxa is cross-arc distance, our goal!

Notice that the cross-track formula relies on the relative bearing, bear13-bear12

We have 3 cases to deal with.

Case 1: The relative bearing is obtuse. So, dxa=dis13.

Case 2.1: The relative bearing is acute, AND p4 falls on our arc. So, dxa=dxt.

Case 2.2: The relative bearing is acute,AND p4 falls beyond our arc. So, dxa=dis23

The algorithm:

Step 1: If relative bearing is obtuse, dxa=dis13
Done!
Step 2: If relative bearing is acute:
2.1: Find dxt.
2.3: Find dis12.
2.4: Find dis14.
2.4: If dis14>dis12, dxa=dis23.
Done!
2.5: If we reach here, dxa=abs(dxt)

MATLAB code:

function [ dxa ] = crossarc( lat1,lon1,lat2,lon2,lat3,lon3 )
%// CROSSARC Calculates the shortest distance in meters 
%// between an arc (defined by p1 and p2) and a third point, p3.
%// Input lat1,lon1,lat2,lon2,lat3,lon3 in degrees.
    lat1=deg2rad(lat1); lat2=deg2rad(lat2); lat3=deg2rad(lat3);
    lon1=deg2rad(lon1); lon2=deg2rad(lon2); lon3=deg2rad(lon3);

    R=6371000; %// Earth's radius in meters
    %// Prerequisites for the formulas
    bear12 = bear(lat1,lon1,lat2,lon2);
    bear13 = bear(lat1,lon1,lat3,lon3);
    dis13 = dis(lat1,lon1,lat3,lon3);

    diff = abs(bear13-bear12);
    if diff > pi
        diff = 2 * pi - diff;
    end
    %// Is relative bearing obtuse?
    if diff>(pi/2)
        dxa=dis13;
    else
        %// Find the cross-track distance.
        dxt = asin( sin(dis13/R)* sin(bear13 - bear12) ) * R;

        %// Is p4 beyond the arc?
        dis12 = dis(lat1,lon1,lat2,lon2);
        dis14 = acos( cos(dis13/R) / cos(dxt/R) ) * R;
        if dis14>dis12
            dxa=dis(lat2,lon2,lat3,lon3);
        else
            dxa=abs(dxt);
        end   
    end
end

function [ d ] = dis( latA, lonA, latB, lonB )
%DIS Finds the distance between two lat/lon points.
R=6371000;
d = acos( sin(latA)*sin(latB) + cos(latA)*cos(latB)*cos(lonB-lonA) ) * R;
end

function [ b ] = bear( latA,lonA,latB,lonB )
%BEAR Finds the bearing from one lat/lon point to another.
b=atan2( sin(lonB-lonA)*cos(latB) , ...
    cos(latA)*sin(latB) - sin(latA)*cos(latB)*cos(lonB-lonA) );
end

Sample outputs: Demonstrate all cases. See maps below.

>> crossarc(-10.1,-55.5,-15.2,-45.1,-10.5,-62.5)
ans =
   7.6709e+05
>> crossarc(40.5,60.5,50.5,80.5,51,69)
ans =
   4.7961e+05
>> crossarc(21.72,35.61,23.65,40.7,25,42)
ans =
   1.9971e+05

Those same outputs on the map!:

Demonstrates case 1:

Demonstrates case 2.1:

Demonstrates case 2.2:

Credit to: http://www.movable-type.co.uk/scripts/latlong.html
for the formulas
and: http://www.darrinward.com/lat-long/?id=1788764
for generating the map images.

Answer 2

Adding a Java version to wdickerson answer:

public static double pointToLineDistance(double lon1, double lat1, double lon2, double lat2, double lon3, double lat3) {
    lat1 = Math.toRadians(lat1);
    lat2 = Math.toRadians(lat2);
    lat3 = Math.toRadians(lat3);
    lon1 = Math.toRadians(lon1);
    lon2 = Math.toRadians(lon2);
    lon3 = Math.toRadians(lon3);

    // Earth's radius in meters
    double R = 6371000;

    // Prerequisites for the formulas
    double bear12 = bear(lat1, lon1, lat2, lon2);
    double bear13 = bear(lat1, lon1, lat3, lon3);
    double dis13 = dis(lat1, lon1, lat3, lon3);

    // Is relative bearing obtuse?
    if (Math.abs(bear13 - bear12) > (Math.PI / 2))
        return dis13;

    // Find the cross-track distance.
    double dxt = Math.asin(Math.sin(dis13 / R) * Math.sin(bear13 - bear12)) * R;

    // Is p4 beyond the arc?
    double dis12 = dis(lat1, lon1, lat2, lon2);
    double dis14 = Math.acos(Math.cos(dis13 / R) / Math.cos(dxt / R)) * R;
    if (dis14 > dis12)
        return dis(lat2, lon2, lat3, lon3);
    return Math.abs(dxt);
}

private static double dis(double latA, double lonA, double latB, double lonB) {
    double R = 6371000;
    return Math.acos(Math.sin(latA) * Math.sin(latB) + Math.cos(latA) * Math.cos(latB) * Math.cos(lonB - lonA)) * R;
}

private static double bear(double latA, double lonA, double latB, double lonB) {
    // BEAR Finds the bearing from one lat / lon point to another.
    return Math.atan2(Math.sin(lonB - lonA) * Math.cos(latB), Math.cos(latA) * Math.sin(latB) - Math.sin(latA) * Math.cos(latB) * Math.cos(lonB - lonA));
}

Answer 3

And adding a python translation of Sga's implementation:

    def bear(latA, lonA, latB, lonB):
        # BEAR Finds the bearing from one lat / lon point to another.
        return math.atan2(
            math.sin(lonB - lonA) * math.cos(latB),
            math.cos(latA) * math.sin(latB) - math.sin(latA) * math.cos(latB) * math.cos(lonB - lonA)
        )


    def pointToLineDistance(lon1, lat1, lon2, lat2, lon3, lat3):
        lat1 = math.radians(lat1)
        lat2 = math.radians(lat2)
        lat3 = math.radians(lat3)
        lon1 = math.radians(lon1)
        lon2 = math.radians(lon2)
        lon3 = math.radians(lon3)
        R = 6378137

        bear12 = bear(lat1, lon1, lat2, lon2)
        bear13 = bear(lat1, lon1, lat3, lon3)
        dis13 = distance( (lat1, lon1), (lat3, lon3)).meters

        # Is relative bearing obtuse?
        if math.fabs(bear13 - bear12) > (math.pi / 2):
            return dis13

        # Find the cross-track distance.
        dxt = math.asin(math.sin(dis13 / R) * math.sin(bear13 - bear12)) * R

        # Is p4 beyond the arc?
        dis12 = distance((lat1, lon1), (lat2, lon2)).meters
        dis14 = math.acos(math.cos(dis13 / R) / math.cos(dxt / R)) * R
        if dis14 > dis12:
            return distance((lat2, lon2), (lat3, lon3)).meters
        return math.fabs(dxt)

Answer 4

For 100 - 1000m spherical problems, it is easy to just convert to cartesian space, using a equirectangular projection.
Then it continues with school mathematics:
Use the function "distance from line segment" which is easy to find ready implemented. This fucntion uses (and sometimes returns) a relative forward/backward position for the projected point X on the line A,B. The value is

• in the interval [0,1] if the projected point is inside the line segment.
• it is negative if X is outside before A,
• it is >1 if outside after B.
  If the relative position is between 0,1 the normal distance is taken, if outside the shorter distance of the both start and line-end points, A,B.

An example of such / or very similar an cartesian implementaion is Shortest distance between a point and a line segment

Answer 5

/**
 * Calculates the euclidean distance from a point to a line segment.
 *
 * @param v     the point
 * @param a     start of line segment
 * @param b     end of line segment 
 * @return      an array of 2 doubles:
 *              [0] distance from v to the closest point of line segment [a,b],
 *              [1] segment coeficient of the closest point of the segment.
 *              Coeficient values < 0 mean the closest point is a.
 *              Coeficient values > 1 mean the closest point is b.
 *              Coeficient values between 0 and 1 mean how far along the segment the closest point is.
 *
 * @author         Afonso Santos
 */
public static
double[]
distanceToSegment( final R3 v, final R3 a, final R3 b )
{
    double[] results    = new double[2] ;

    final R3     ab_    = b.sub( a ) ;
    final double ab     = ab_.modulus( ) ;

    final R3     av_    = v.sub( a ) ;
    final double av     = av_.modulus( ) ;

    if (ab == 0.0)                       // a and b coincide
    {
        results[0] = av ;                // Distance
        results[1] = 0.0 ;               // Segment coeficient.
    }
    else
    {
        final double avScaProjAb  = av_.dot(ab_) / ab ;
        final double abCoeficient = results[1] = avScaProjAb / ab ;

        if (abCoeficient <= 0.0)                 // Point is before start of the segment ?
            results[0] = av ;                    // Use distance to start of segment.
        else if (abCoeficient >= 1.0)            // Point is past the end of the segment ?
            results[0] = v.sub( b ).modulus() ;    // Use distance to end of segment.
        else                                       // Point is within the segment's start/end perpendicular boundaries.
        {
            if (avScaProjAb >= av)                    // Test to avoid machine float representation epsilon rounding errors that would result in expection on sqrt.
                results[0] = 0.0 ;                    // a, b and v are colinear.
            else
                results[0] = Math.sqrt( av * av - avScaProjAb * avScaProjAb ) ;        // Perpendicular distance from point to segment.
        }
    }

    return results ;
}

the above method requires cartesian 3D space arguments and you asked to use lat/lon arguments. To do the conversion use

/**
 * Calculate 3D vector (from center of earth).
 * 
 * @param latDeg    latitude (degrees)
 * @param lonDeg    longitude (degrees)
 * @param eleMtr    elevation (meters)
 * @return          3D cartesian vector (from center of earth).
 * 
 * @author          Afonso Santos
 */
public static
R3
cartesian( final double latDeg, final double lonDeg, final double eleMtr )
{
    return versor( latDeg, lonDeg ).scalar( EARTHMEANRADIUS_MTR + eleMtr ) ;
}

For the rest of the 3D/R3 code or how to calculate distance to a path/route/track check https://sourceforge.net/projects/geokarambola/

Answer 6

Adding an ObjectiveC translation of wdickerson implementation:

#define DEGREES_RADIANS(angle) ((angle) / 180.0 * M_PI)
#define RADIANS_DEGREES(angle) ((angle) / M_PI * 180)

- (double)crossArcFromCoord:(CLLocationCoordinate2D)fromCoord usingArcFromCoord:(CLLocationCoordinate2D)arcCoord1 toArcCoord:(CLLocationCoordinate2D)arcCoord2 {

        fromCoord.latitude = DEGREES_RADIANS(fromCoord.latitude);
        fromCoord.longitude = DEGREES_RADIANS(fromCoord.longitude);

        arcCoord1.latitude = DEGREES_RADIANS(arcCoord1.latitude);
        arcCoord1.longitude = DEGREES_RADIANS(arcCoord1.longitude);

        arcCoord2.latitude = DEGREES_RADIANS(arcCoord2.latitude);
        arcCoord2.longitude = DEGREES_RADIANS(arcCoord2.longitude);

        double R = 6371000; // Earth's radius in meters

        // Prerequisites for the formulas
        double bear12 = [self bearFromCoord:arcCoord1 toCoord:arcCoord2];
        double bear13 = [self bearFromCoord:arcCoord1 toCoord:fromCoord];

        double dis13 = [self distFromCoord:arcCoord1 toCoord:fromCoord];

        double diff = fabs(bear13 - bear12);

        if (diff > M_PI) {
            diff = 2 * M_PI - diff;
        }

        // Is relative bearing obtuse?
        if (diff > (M_PI/2)) {
            return dis13;
        }

        // Find the cross-track distance
        double dxt = asin(sin(dis13 / R) * sin(bear13 - bear12)) * R;

        // Is p4 beyond the arc?
        double dis12 = [self distFromCoord:arcCoord1 toCoord:arcCoord2];
        double dis14 = acos(cos(dis13 / R) / cos(dxt / R)) * R;

        if (dis14 > dis12) {
            return [self distFromCoord:arcCoord2 toCoord:fromCoord];
        }

        return fabs(dxt);
    }

    - (double)distFromCoord:(CLLocationCoordinate2D)coord1 toCoord:(CLLocationCoordinate2D)coord2 {

        double R = 6371000;

        return acos(sin(coord1.latitude) * sin(coord2.latitude) + cos(coord1.latitude) * cos(coord2.latitude) * cos(coord2.longitude - coord2.longitude)) * R;
    }

    - (double)bearFromCoord:(CLLocationCoordinate2D)fromCoord toCoord:(CLLocationCoordinate2D)toCoord {

        return atan2(sin(toCoord.longitude - fromCoord.longitude) * cos(toCoord.latitude),
                     cos(fromCoord.latitude) * sin(toCoord.latitude) - (sin(fromCoord.latitude) * cos(toCoord.latitude) * cos(toCoord.longitude - fromCoord.longitude)));
     }

Answer 7

Adding a python+numpy implementation (now you can pass your longitudes and latitudes as arrays and compute all your distances simultaneously without loops).

def _angularSeparation(long1, lat1, long2, lat2):
    """All radians
    """
    t1 = np.sin(lat2/2.0 - lat1/2.0)**2
    t2 = np.cos(lat1)*np.cos(lat2)*np.sin(long2/2.0 - long1/2.0)**2
    _sum = t1 + t2

    if np.size(_sum) == 1:
        if _sum < 0.0:
            _sum = 0.0
    else:
        _sum = np.where(_sum < 0.0, 0.0, _sum)

    return 2.0*np.arcsin(np.sqrt(_sum))


def bear(latA, lonA, latB, lonB):
    """All radians
    """
    # BEAR Finds the bearing from one lat / lon point to another.
    result = np.arctan2(np.sin(lonB - lonA) * np.cos(latB),
                        np.cos(latA) * np.sin(latB) - np.sin(latA) * np.cos(latB) * np.cos(lonB - lonA)
                        )

    return result


def pointToLineDistance(lon1, lat1, lon2, lat2, lon3, lat3):
    """All radians
    points 1 and 2 define an arc segment,
    this finds the distance of point 3 to the arc segment. 
    """

    result = lon1*0
    needed = np.ones(result.size, dtype=bool)

    bear12 = bear(lat1, lon1, lat2, lon2)
    bear13 = bear(lat1, lon1, lat3, lon3)
    dis13 = _angularSeparation(lon1, lat1, lon3, lat3)

    # Is relative bearing obtuse?
    diff = np.abs(bear13 - bear12)
    if np.size(diff) == 1:
        if diff > np.pi:
            diff = 2*np.pi - diff
        if diff > (np.pi / 2):
            return dis13
    else:
        solved = np.where(diff > (np.pi / 2))[0]
        result[solved] = dis13[solved]
        needed[solved] = 0
    
    # Find the cross-track distance.
    dxt = np.arcsin(np.sin(dis13) * np.sin(bear13 - bear12))

    # Is p4 beyond the arc?
    dis12 = _angularSeparation(lon1, lat1, lon2, lat2)
    dis14 = np.arccos(np.cos(dis13) / np.cos(dxt))
    if np.size(dis14) == 1:
        if dis14 > dis12:
            return _angularSeparation(lon2, lat2, lon3, lat3)
    else:
        solved = np.where(dis14 > dis12)[0]
        result[solved] = _angularSeparation(lon2[solved], lat2[solved], lon3[solved], lat3[solved])

    if np.size(lon1) == 1:
        return np.abs(dxt)
    else:
        result[needed] = np.abs(dxt[needed])
        return result