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CLLocation+Navigation.m
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CLLocation+Navigation.m
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//
// CLLocation+Navigation.m
// Koolistov
//
// Created by Johan Kool on 22-11-10.
// Copyright 2010-2011 Koolistov. All rights reserved.
//
// Redistribution and use in source and binary forms, with or without modification, are
// permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice, this list of
// conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice, this list
// of conditions and the following disclaimer in the documentation and/or other materials
// provided with the distribution.
// * Neither the name of KOOLISTOV nor the names of its contributors may be used to
// endorse or promote products derived from this software without specific prior written
// permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
// MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL
// THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT
// OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
// HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
// OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
//
// This is an adaptation of the work done by Chris Veness (Attribution 3.0 Unported (CC BY 3.0)):
// Latitude/longitude spherical geodesy formulae & scripts (c) Chris Veness 2002-2010
// http://www.movable-type.co.uk/scripts/latlong.html
//
#import "CLLocation+Navigation.h"
#define degreesToRadians(x) (M_PI * x / 180.0)
#define radiansToDegrees(x) (x * (180.0 / M_PI))
#define EARTH_RADIUS 6371009.0; // Earth radius in meters (same unit as d) (Using mean radius as defined on Wikipedia)
@implementation CLLocation (Navigation)
#pragma mark Circle Paths
- (CLLocationDistance)kv_distanceUsingHaversineAlongCirclePathToCoordinate:(CLLocationCoordinate2D)destination {
// Haversine formula:
//
// R = earth’s radius (mean radius = 6,371km)
// Δlat = lat2− lat1
// Δlong = long2− long1
// a = sin²(Δlat/2) + cos(lat1).cos(lat2).sin²(Δlong/2)
// c = 2.atan2(√a, √(1−a))
// d = R.c
//
// (Note that angles need to be in radians to pass to trig functions).
//
// JavaScript:
// var R = 6371; // km
// var dLat = (lat2-lat1).toRad();
// var dLon = (lon2-lon1).toRad();
// var a = Math.sin(dLat/2) * Math.sin(dLat/2) +
// Math.cos(lat1.toRad()) * Math.cos(lat2.toRad()) *
// Math.sin(dLon/2) * Math.sin(dLon/2);
// var c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1-a));
// var d = R * c;
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double lat2 = degreesToRadians(destination.latitude);
double lon2 = degreesToRadians(destination.longitude);
double R = EARTH_RADIUS;
double dLat = lat2 - lat1;
double dLon = lon2 - lon1;
double a = sin(dLat / 2.0) * sin(dLat / 2.0) + cos(lat1) * cos(lat2) * sin(dLon / 2.0) * sin(dLon / 2.0);
double c = 2.0 * atan2(sqrt(a), sqrt(1.0 - a));
double d = R * c;
return d;
}
- (CLLocationDistance)kv_distanceUsingSphericalLawOfCosinesAlongCirclePathToCoordinate:(CLLocationCoordinate2D)destination {
// Spherical law
// of cosines: d = acos(sin(lat1).sin(lat2)+cos(lat1).cos(lat2).cos(long2−long1)).R
// JavaScript:
// var R = 6371; // km
// var d = Math.acos(Math.sin(lat1)*Math.sin(lat2) +
// Math.cos(lat1)*Math.cos(lat2) *
// Math.cos(lon2-lon1)) * R;
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double lat2 = degreesToRadians(destination.latitude);
double lon2 = degreesToRadians(destination.longitude);
double R = EARTH_RADIUS;
double dLon = lon2 - lon1;
double d = acos(sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(dLon)) * R;
return d;
}
- (CLLocationDirection)kv_intialBearingOnCirclePathToCoordinate:(CLLocationCoordinate2D)destination {
// Formula: θ = atan2( sin(Δlong).cos(lat2),
// cos(lat1).sin(lat2) − sin(lat1).cos(lat2).cos(Δlong) )
// JavaScript:
// var y = Math.sin(dLon) * Math.cos(lat2);
// var x = Math.cos(lat1)*Math.sin(lat2) -
// Math.sin(lat1)*Math.cos(lat2)*Math.cos(dLon);
// var brng = Math.atan2(y, x).toDeg();
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double lat2 = degreesToRadians(destination.latitude);
double lon2 = degreesToRadians(destination.longitude);
double dLon = lon2 - lon1;
double y = sin(dLon) * cos(lat2);
double x = cos(lat1) * sin(lat2) - sin(lat1) * cos(lat2) * cos(dLon);
double bearing = atan2(y, x);
return fmod((radiansToDegrees(bearing) + 360.0), 360.0);
}
- (CLLocationDirection)kv_finalBearingOnCirclePathToCoordinate:(CLLocationCoordinate2D)destination {
// Calculate initial bearing of reverse path
double lat1 = degreesToRadians(destination.latitude);
double lon1 = degreesToRadians(destination.longitude);
double lat2 = degreesToRadians(self.coordinate.latitude);
double lon2 = degreesToRadians(self.coordinate.longitude);
double dLon = lon2 - lon1;
double y = sin(dLon) * cos(lat2);
double x = cos(lat1) * sin(lat2) - sin(lat1) * cos(lat2) * cos(dLon);
double bearing = atan2(y, x);
double initialBearingReversePath = fmod((radiansToDegrees(bearing) + 360.0), 360.0);
// Flip result
return fmod((initialBearingReversePath + 180.0), 360.0);
}
- (CLLocationCoordinate2D)kv_midPointOnCirclePathToCoordinate:(CLLocationCoordinate2D)destination {
// Formula: Bx = cos(lat2).cos(Δlong)
// By = cos(lat2).sin(Δlong)
// latm = atan2(sin(lat1) + sin(lat2), √((cos(lat1)+Bx)² + By²))
// lonm = lon1 + atan2(By, cos(lat1)+Bx)
// JavaScript:
// var Bx = Math.cos(lat2) * Math.cos(dLon);
// var By = Math.cos(lat2) * Math.sin(dLon);
// var lat3 = Math.atan2(Math.sin(lat1)+Math.sin(lat2),
// Math.sqrt( (Math.cos(lat1)+Bx)*(Math.cos(lat1)+Bx) + By*By) );
// var lon3 = lon1 + Math.atan2(By, Math.cos(lat1) + Bx);
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double lat2 = degreesToRadians(destination.latitude);
double lon2 = degreesToRadians(destination.longitude);
double dLon = lon2 - lon1;
double x = cos(lat2) * cos(dLon);
double y = cos(lat2) * sin(dLon);
double lat3 = atan2(sin(lat1) + sin(lat2), sqrt((cos(lat1) + x) * (cos(lat1) + x) + y * y));
double lon3 = lon1 + atan2(y, cos(lat1) + x);
CLLocationCoordinate2D midPoint = CLLocationCoordinate2DMake(radiansToDegrees(lat3), radiansToDegrees(lon3));
return midPoint;
}
- (CLLocationCoordinate2D)kv_destinationCoordinateOnCirclePathUsingInitialBearing:(CLLocationDirection)initialBearing andDistance:(CLLocationDistance)distance {
// Formula: lat2 = asin(sin(lat1)*cos(d/R) + cos(lat1)*sin(d/R)*cos(θ))
// lon2 = lon1 + atan2(sin(θ)*sin(d/R)*cos(lat1), cos(d/R)−sin(lat1)*sin(lat2))
// d/R is the angular distance (in radians), where d is the distance travelled and R is the earth’s radius
// JavaScript:
// var lat2 = Math.asin( Math.sin(lat1)*Math.cos(d/R) +
// Math.cos(lat1)*Math.sin(d/R)*Math.cos(brng) );
// var lon2 = lon1 + Math.atan2(Math.sin(brng)*Math.sin(d/R)*Math.cos(lat1),
// Math.cos(d/R)-Math.sin(lat1)*Math.sin(lat2));
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double brng = degreesToRadians(initialBearing);
double d = distance;
double R = EARTH_RADIUS;
double lat2 = asin(sin(lat1) * cos(d / R) + cos(lat1) * sin(d / R) * cos(brng));
double lon2 = lon1 + atan2(sin(brng) * sin(d / R) * cos(lat1), cos(d / R) - sin(lat1) * sin(lat2));
CLLocationCoordinate2D destination = CLLocationCoordinate2DMake(radiansToDegrees(lat2), radiansToDegrees(lon2));
return destination;
}
// Intersection of two paths given start points and bearings.
- (CLLocationCoordinate2D)kv_intersectionCoordinateForCirclePathWithBearing:(CLLocationDirection)initialBearing1 andCirclePathFromCoordinate:(CLLocationCoordinate2D)departure2 bearing:(CLLocationDirection)initialBearing2 {
// LatLon.intersection = function(p1, brng1, p2, brng2) {
// brng1 = typeof brng1 == 'number' ? brng1 : typeof brng1 == 'string' && trim(brng1)!='' ? +brng1 : NaN;
// brng2 = typeof brng2 == 'number' ? brng2 : typeof brng2 == 'string' && trim(brng2)!='' ? +brng2 : NaN;
// lat1 = p1._lat.toRad(), lon1 = p1._lon.toRad();
// lat2 = p2._lat.toRad(), lon2 = p2._lon.toRad();
// brng13 = brng1.toRad(), brng23 = brng2.toRad();
// dLat = lat2-lat1, dLon = lon2-lon1;
//
// dist12 = 2*Math.asin( Math.sqrt( Math.sin(dLat/2)*Math.sin(dLat/2) +
// Math.cos(lat1)*Math.cos(lat2)*Math.sin(dLon/2)*Math.sin(dLon/2) ) );
// if (dist12 == 0) return null;
//
// // initial/final bearings between points
// brngA = Math.acos( ( Math.sin(lat2) - Math.sin(lat1)*Math.cos(dist12) ) /
// ( Math.sin(dist12)*Math.cos(lat1) ) );
// if (isNaN(brngA)) brngA = 0; // protect against rounding
// brngB = Math.acos( ( Math.sin(lat1) - Math.sin(lat2)*Math.cos(dist12) ) /
// ( Math.sin(dist12)*Math.cos(lat2) ) );
//
// if (Math.sin(lon2-lon1) > 0) {
// brng12 = brngA;
// brng21 = 2*Math.PI - brngB;
// } else {
// brng12 = 2*Math.PI - brngA;
// brng21 = brngB;
// }
//
// alpha1 = (brng13 - brng12 + Math.PI) % (2*Math.PI) - Math.PI; // angle 2-1-3
// alpha2 = (brng21 - brng23 + Math.PI) % (2*Math.PI) - Math.PI; // angle 1-2-3
//
// if (Math.sin(alpha1)==0 && Math.sin(alpha2)==0) return null; // infinite intersections
// if (Math.sin(alpha1)*Math.sin(alpha2) < 0) return null; // ambiguous intersection
//
// //alpha1 = Math.abs(alpha1);
// //alpha2 = Math.abs(alpha2);
// // ... Ed Williams takes abs of alpha1/alpha2, but seems to break calculation?
//
// alpha3 = Math.acos( -Math.cos(alpha1)*Math.cos(alpha2) +
// Math.sin(alpha1)*Math.sin(alpha2)*Math.cos(dist12) );
// dist13 = Math.atan2( Math.sin(dist12)*Math.sin(alpha1)*Math.sin(alpha2),
// Math.cos(alpha2)+Math.cos(alpha1)*Math.cos(alpha3) )
// lat3 = Math.asin( Math.sin(lat1)*Math.cos(dist13) +
// Math.cos(lat1)*Math.sin(dist13)*Math.cos(brng13) );
// dLon13 = Math.atan2( Math.sin(brng13)*Math.sin(dist13)*Math.cos(lat1),
// Math.cos(dist13)-Math.sin(lat1)*Math.sin(lat3) );
// lon3 = lon1+dLon13;
// lon3 = (lon3+Math.PI) % (2*Math.PI) - Math.PI; // normalise to -180..180º
//
// return new LatLon(lat3.toDeg(), lon3.toDeg());
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double lat2 = degreesToRadians(departure2.latitude);
double lon2 = degreesToRadians(departure2.longitude);
double brng13 = degreesToRadians(initialBearing1);
double brng23 = degreesToRadians(initialBearing2);
double dLat = lat2 - lat1;
double dLon = lon2 - lon1;
double dist12 = 2.0 * asin(sqrt(sin(dLat / 2.0) * sin(dLat / 2.0) + cos(lat1) * cos(lat2) * sin(dLon / 2.0) * sin(dLon / 2.0)));
if (dist12 == 0.0) return self.coordinate; // same coordinates so return any of the coordinates
double brngA = acos((sin(lat2) - sin(lat1) * cos(dist12)) / (sin(dist12) * cos(lat1)));
if (isnan(brngA)) brngA = 0.0;
double brngB = acos( ( sin(lat1) - sin(lat2) * cos(dist12) ) / ( sin(dist12) * cos(lat2) ) );
double brng12, brng21;
if (sin(dLon) > 0.0) {
brng12 = brngA;
brng21 = 2.0 * M_PI - brngB;
} else {
brng12 = 2.0 * M_PI - brngA;
brng21 = brngB;
}
double alpha1 = fmod((brng13 - brng12 + M_PI), (2.0 * M_PI)) - M_PI; // angle 2-1-3
double alpha2 = fmod((brng21 - brng23 + M_PI), (2.0 * M_PI)) - M_PI; // angle 1-2-3
if (sin(alpha1) == 0.0 && sin(alpha2) == 0.0) return CLLocationCoordinate2DMake(-91.0, -181.0); // return invalid coordinate: infinite intersections
if (sin(alpha1) * sin(alpha2) < 0.0) return CLLocationCoordinate2DMake(-91.0, -181.0); // return invalid coordinate: ambiguous intersection
double alpha3 = acos(-cos(alpha1) * cos(alpha2) +
sin(alpha1) * sin(alpha2) * cos(dist12) );
double dist13 = atan2(sin(dist12) * sin(alpha1) * sin(alpha2),
cos(alpha2) + cos(alpha1) * cos(alpha3) );
double lat3 = asin(sin(lat1) * cos(dist13) +
cos(lat1) * sin(dist13) * cos(brng13) );
double dLon13 = atan2(sin(brng13) * sin(dist13) * cos(lat1),
cos(dist13) - sin(lat1) * sin(lat3) );
double lon3 = lon1 + dLon13;
lon3 = fmod((lon3 + M_PI), (2.0 * M_PI)) - M_PI; // normalise to -180..180º
CLLocationCoordinate2D intersection = CLLocationCoordinate2DMake(radiansToDegrees(lat3), radiansToDegrees(lon3));
return intersection;
}
// - (CLLocationDistance)kv_distanceFromCirclePathFromCoordinate:(CLLocationCoordinate2D)departure toCoordinate:(CLLocationCoordinate2D)destination {
// // Formula: dxt = asin(sin(d13/R)*sin(θ13−θ12)) * R
// // where d13 is 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
// // JavaScript:
// // var dXt = Math.asin(Math.sin(d13/R)*Math.sin(brng13-brng12)) * R;
//
// double dXt = asin(sin(d13/R)*sin(brng13-brng12)) * R;
// }
- (CLLocationDegrees)kv_maximumLatitudeOfCirclePathToCoordinate:(CLLocationCoordinate2D)destination {
// Formula: latmax = acos(abs(sin(θ)*cos(lat)))
// JavaScript:
// var latMax = Math.acos(Math.abs(Math.sin(brng)*Math.cos(lat)));
double bearing = [self kv_intialBearingOnCirclePathToCoordinate:destination];
double brng = degreesToRadians(bearing);
double lat = degreesToRadians(self.coordinate.latitude);
double latMax = acos(abs(sin(brng) * cos(lat)));
return radiansToDegrees(latMax);
}
#pragma mark -
#pragma mark Rhumb Lines
- (CLLocationDirection)kv_bearingOnRhumbLineToCoordinate:(CLLocationCoordinate2D)destination {
// Formula: Δφ = ln(tan(lat2/2+π/4)/tan(lat1/2+π/4)) [= the ‘stretched’ latitude difference]
// if E:W line, q = cos(lat1)
// otherwise, q = Δlat/Δφ
// d = √(Δlat² + q².Δlon²).R [pythagoras]
// θ = atan2(Δlon, Δφ)
// where ln is natural log, Δlon is taking shortest route (<180º), and R is the earth’s radius
// JavaScript:
// var dPhi = Math.log(Math.tan(lat2/2+Math.PI/4)/Math.tan(lat1/2+Math.PI/4));
// var q = (!isNaN(dLat/dPhi)) ? dLat/dPhi : Math.cos(lat1); // E-W line gives dPhi=0
//
// // if dLon over 180° take shorter rhumb across 180° meridian:
// if (Math.abs(dLon) > Math.PI) {
// dLon = dLon>0 ? -(2*Math.PI-dLon) : (2*Math.PI+dLon);
// }
// var d = Math.sqrt(dLat*dLat + q*q*dLon*dLon) * R;
// var brng = Math.atan2(dLon, dPhi);
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double lat2 = degreesToRadians(destination.latitude);
double lon2 = degreesToRadians(destination.longitude);
double dLon = lon2 - lon1;
double dPhi = log(tan(lat2 / 2 + M_PI_4) / tan(lat1 / 2 + M_PI_4));
if (abs(dLon) > M_PI) {
dLon = (dLon > 0.0) ? -(2.0 * M_PI - dLon) : (2.0 * M_PI + dLon);
}
double brng = atan2(dLon, dPhi);
return fmod((radiansToDegrees(brng) + 360.0), 360.0);
}
- (CLLocationDirection)kv_distanceAlongRhumbLineToCoordinate:(CLLocationCoordinate2D)destination {
// Formula: Δφ = ln(tan(lat2/2+π/4)/tan(lat1/2+π/4)) [= the ‘stretched’ latitude difference]
// if E:W line, q = cos(lat1)
// otherwise, q = Δlat/Δφ
// d = √(Δlat² + q².Δlon²).R [pythagoras]
// θ = atan2(Δlon, Δφ)
// where ln is natural log, Δlon is taking shortest route (<180º), and R is the earth’s radius
// JavaScript:
// var dPhi = Math.log(Math.tan(lat2/2+Math.PI/4)/Math.tan(lat1/2+Math.PI/4));
// var q = (!isNaN(dLat/dPhi)) ? dLat/dPhi : Math.cos(lat1); // E-W line gives dPhi=0
//
// // if dLon over 180° take shorter rhumb across 180° meridian:
// if (Math.abs(dLon) > Math.PI) {
// dLon = dLon>0 ? -(2*Math.PI-dLon) : (2*Math.PI+dLon);
// }
// var d = Math.sqrt(dLat*dLat + q*q*dLon*dLon) * R;
// var brng = Math.atan2(dLon, dPhi);
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double lat2 = degreesToRadians(destination.latitude);
double lon2 = degreesToRadians(destination.longitude);
double R = EARTH_RADIUS;
double dLat = lat2 - lat1;
double dLon = lon2 - lon1;
double dPhi = log(tan(lat2 / 2 + M_PI_4) / tan(lat1 / 2 + M_PI_4));
double q = (!isnan(dLat / dPhi) && !isinf(dLat / dPhi)) ? dLat / dPhi : cos(lat1);
if (abs(dLon) > M_PI) {
dLon = (dLon > 0.0) ? -(2.0 * M_PI - dLon) : (2.0 * M_PI + dLon);
}
double d = sqrt(dLat * dLat + q * q * dLon * dLon) * R;
return d;
}
- (CLLocationCoordinate2D)kv_destinationCoordinateOnRhumbLineUsingBearing:(CLLocationDirection)bearing andDistance:(CLLocationDistance)distance {
// Formula: α = d/R (angular distance)
// lat2 = lat1 + α.cos(θ)
// Δφ = ln(tan(lat2/2+π/4)/tan(lat1/2+π/4)) [= the ‘stretched’ latitude difference]
// if E:W line q = cos(lat1)
// otherwise q = Δlat/Δφ
// Δlon = α.sin(θ)/q
// lon2 = (lon1+Δlon+π) % 2.π − π
// where ln is natural log and % is modulo, Δlon is taking shortest route (<180°), and R is the earth’s radius
// JavaScript:
// lat2 = lat1 + d*Math.cos(brng);
// var dPhi = Math.log(Math.tan(lat2/2+Math.PI/4)/Math.tan(lat1/2+Math.PI/4));
// var q = (!isNaN(dLat/dPhi)) ? dLat/dPhi : Math.cos(lat1); // E-W line gives dPhi=0
//
// var dLon = d*Math.sin(brng)/q;
// // check for some daft bugger going past the pole, normalise latitude if so
// if (Math.abs(lat2) > Math.PI/2) lat2 = lat2>0 ? Math.PI-lat2 : -(Math.PI-lat2);
// lon2 = (lon1+dLon+Math.PI)%(2*Math.PI) - Math.PI;
double lat1 = degreesToRadians(self.coordinate.latitude);
double lon1 = degreesToRadians(self.coordinate.longitude);
double brng = degreesToRadians(bearing);
double d = distance;
double R = EARTH_RADIUS;
double lat2 = lat1 + (d / R) * cos(brng);
double dLat = lat2 - lat1;
double dPhi = log(tan(lat2 / 2 + M_PI_4) / tan(lat1 / 2 + M_PI_4));
double q = (!isnan(dLat / dPhi) && !isinf(dLat / dPhi)) ? dLat / dPhi : cos(lat1);
double dLon = (d / R) * sin(brng) / q;
if (abs(lat2) > M_PI_2) {
lat2 = (lat2 > 0) ? M_PI - lat2 : -(M_PI - lat2);
}
double lon2 = fmod((lon1 + dLon + 3 * M_PI), 2.0 * M_PI) - M_PI;
CLLocationCoordinate2D destination = CLLocationCoordinate2DMake(radiansToDegrees(lat2), radiansToDegrees(lon2));
return destination;
}
@end