I am trying to calculate bearing between two lat/lon points as given in this link. I see that the bearing we get initially using the below equation is initial bearing.
public static double GetBearing(double latitude1, double longitude1, double latitude2, double longitude2)
{
var lat1 = ToRadians(latitude1);
var lat2 = ToRadians(latitude2);
var longdiff = ToRadians(longitude1 - longitude2);
var X = Math.Cos(lat2) * Math.Sin(longdiff);
var Y = Math.Cos(lat1) * Math.Sin(lat2) - Math.Sin(lat1) * Math.Cos(lat2) * Math.Cos(longdiff);
var bearing =ToDegrees(Math.Atan2(X, Y));
return (bearing+360)%360;
}
It is given that
For final bearing, simply take the initial bearing from the end point to the start point and reverse it (using θ = (θ+180) % 360).
I am confused about the difference between initial bearing and final bearing.
What is this initial and final bearing and which bearing should we take as the final answer for bearing between two points.
The bearing is the angle between direction along the shortest path to destination and direction to North. The reason we have initial and final one is that we live on sphere, so the shortest path is geodesic line. It is a straight line on globe, bit if you draw it on flat map - it will be a curve.
There are two ways to think about it. Thinking on flat map: as you travel from A to B, this curve changes direction slightly, so the angle between this line and North changes, i.e. bearing changes.
Or you can think on sphere, and then think about triangle A - B - North Pole. The bearing is angle between between AB and appropriate meridian. Initial bearing is angle between AB and meridian crossing A. Final one is angle between AB and meridian crossing B. They are different.
The single "final answer" bearing only makes sense when distance between A and B is short. Then the curvature of Earth does not matter much, and the initial and final bearings are very close to each other, so depending on precision needed one can talk about single bearing.
FYI: bearing and many related computations are implemented in the R package geosphere
The bearing function returns the initial bearing, but you can invert the coordinates to get the final bearing.
library(geosphere)
bearing(cbind(0,0),cbind(20,20))
#[1] 43.4035
finalb <- bearing(cbind(20,20),cbind(0,0))
(finalb + 180) %% 360
#[1] 46.9656
(these results should be more precise than the ones you get with algorithm you refer to)
def bearing(lat1, lon1, lat2, lon2):
# Convert latitude and longitude to radians
lat1 = math.radians(lat1)
lon1 = math.radians(lon1)
lat2 = math.radians(lat2)
lon2 = math.radians(lon2)
y = math.sin(lon2-lon1) * math.cos(lat2)
x = math.cos(lat1)*math.sin(lat2) - math.sin(lat1)*math.cos(lat2)*math.cos(lon2-lon1)
initial_bearing = math.degrees(math.atan2(y, x))
final_bearing = (initial_bearing + 180) % 360 if initial_bearing < 180 else (initial_bearing - 180) % 360
return initial_bearing, final_bearing
I have pairs of gps coordinates (longitude latitude) and I would like to calculate the walking distance between them. i.e. using road data (from google maps or another open source) calculate the km of the shortest route between the two gps points. I could do it using google maps, but I have thousands of pairs so I would like to find a more automated way.
Does somebody know how to do it?
I am not quite sure about what you looking for. Just share some thoughts here:
1) If you want to calculate great circle distance between two points in lat/lon, you could use haversine formula distance. Example in JS:
function Haversine_distance(lat1,lon1,lat2,lon2) {
var R = 6371; // in km
var x1 = lat2 - lat1;
var dLat = x1.toRad();
var x2 = lon2 - lon1;
var dLon = x2.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;
return d*1.0;
};
2) If you need more accurate distance calculation you need some correction factor, since earth is not a perfer sphere. It is always easier to project the locations you have to an appropriate projection and calculate distance there. For instance, project to UTM zones using proj4js, then calculate the distance to reduce the inaccuracy.
3) If you are talking about walking distance in cities, then it is network distance. It is required to have your road network build up first, then calculate from there. Without the road network, giving only point locations will not be enough to calculate the walking distance. Commercial data for road network is available from such as TeleAtlas. Free data can also be found via OpenStreetMap.
I'm trying to find a function lng = f(lat) that would help me draw a line between 2 given GPS coordinates, (lat1, lng1) and (lat2, lng2).
I've tried the traditional Cartesian formula y=mx+b where m=(y2-y1)/(x2-x1), but GPS coordinates don't seem to behave that way.
What would be a formula/algorithm that could help me achieve my goal.
PS: I'm using Google Maps API but let's keep this implementation agnostic if possible.
UPDATE: My implementation was wrong and it seems the algorithm is actually working as stated by some of the answers. My bad :(
What you want to do should actually work. Keep in mind however that if north is on top, the horizontal (x) axis is the LONGITUDE and the vertical (y) axis is the LATITUDE (I think you might have confused this).
If you parametrize the line as lat = func(long) you will run into trouble with vertical lines (i.e. those going exactly north south) as the latitude varies while the longitude is fixed.
Therefore I'd rather use another parametrization:
long(alpha) = long_1 + alpha * (long_2 - long_1)
lat(alpha) = lat_1 + alpha * (lat_2 - lat_1)
and vary alpha from 0 to 1.
This will not exactly coincide with a great circle (shortest path on a sphere) but the smaller the region you are looking at, the less noticeable the difference will be (as others posters here pointed out).
Here is a distance formula I use that may help. This is using javascript.
function Distance(lat1, lat2, lon1, lon2) {
var R = 6371; // km
var dLat = toRad(lat2 - lat1);
var dLon = toRad(lon2 - lon1);
var a = Math.sin(dLat / 2) * Math.sin(dLat / 2) + Math.cos(toRad(lat1)) * Math.cos(toRad(lat2)) * Math.sin(dLon / 2) * Math.sin(dLon / 2);
var c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1 - a));
var d = R * c * 0.621371;
var r = Math.round(d * 100) / 100;
return r;
}
For short distances, where the earth curvature doesn't make a significant difference, it works fine to draw a line using regular two-dimensional geometry.
For longer distances the shortest way between two lines does not project as a straight line on a map, but as a curve. (For example, the shortest way from Sweden to Alaska would be straight over the noth pole, not past Canada and Iceland.) You would have to use three-dimensional geometry to draw a line on a surface of a sphere, then project that onto the map in the same way the earth surface is projected on the map.
Is your goal to find this equation or to actually draw a line?
If the latter, since you're using the Maps API, specify geodesic: true and draw it with a Polyline:
http://code.google.com/apis/maps/documentation/javascript/reference.html#Polyline
Suppose I have the following:
A region defined by minimum and maximum latitude and longitude (commonly a 'lat-long rect', though it's not actually rectangular except in certain projections).
A circle, defined by a center lat/long and a radius
How can I determine:
Whether the two shapes overlap?
Whether the circle is entirely contained within the rect?
I'm looking for a complete formula/algorithm, rather than a lesson in the math, per-se.
warning: this can be tricky if the circles / "rectangles" span large portions of the sphere, e.g.:
"rectangle": min long = -90deg, max long = +90deg, min lat = +70deg, max lat = +80deg
circle: center = lat = +85deg, long = +160deg, radius = 20deg (e.g. if point A is on the circle and point C is the circle's center, and point O is the sphere's center, then angle AOC = 40deg).
These intersect but the math is likely to have several cases to check intersection/containment. The following points lie on the circle described above: P1=(+65deg lat,+160deg long), P2=(+75deg lat, -20deg long). P1 is outside the "rectangle" and P2 is inside the "rectangle" so the circle/"rectangle" intersect in at least 2 points.
OK, here's my shot at an outline of the solution:
Let C = circle center with radius R (expressed as a spherical angle as above). C has latitude LATC and longitude LONGC. Since the word "rectangle" is kind of misleading here (lines of constant latitude are not segments of great circles), I'll use the term "bounding box".
function InsideCircle(P) returns +1,0,or -1: +1 if point P is inside the circle, 0 if point P is on the circle, and -1 if point P is outside the circle: calculation of great-circle distance D (expressed as spherical angle) between C and any point P will tell you whether or not P is inside the circle: InsideCircle(P) = sign(R-D) (As user #Die in Sente mentioned, great circle distances have been asked on this forum elsewhere)
Define PANG(x) = the principal angle of x = MOD(x+180deg, 360deg)-180deg. PANG(x) is always between -180deg and +180deg, inclusive (+180deg should map to -180deg).
To define the bounding box, you need to know 4 numbers, but there's a slight issue with longitude. LAT1 and LAT2 represent the bounding latitudes (assuming LAT1 < LAT2); there's no ambiguity there. LONG1 and LONG2 represent the bounding longitudes of a longitude interval, but this gets tricky, and it's easier to rewrite this interval as a center and width, with LONGM = the center of that interval and LONGW = width. NOTE that there are always 2 possibilities for longitude intervals. You have to specify which of these cases it is, whether you are including or excluding the 180deg meridian, e.g. the shortest interval from -179deg to +177deg has LONGM = +179deg and LONGW = 4deg, but the other interval from -179deg to +177deg has LONGM = -1deg and LONGW = 356deg. If you naively try to do "regular" comparisons with the interval [-179,177] you will end up using the larger interval and that's probably not what you want. As an aside, point P, with latitude LATP and longitude LONGP, is inside the bounding box if both of the following are true:
LAT1 <= LATP and LATP <= LAT2 (that part is obvious)
abs(PANG(LONGP-LONGM)) < LONGW/2
The circle intersects the bounding box if ANY of the following points P in PTEST = union(PCORNER,PLAT,PLONG) as described below, do not all return the same result for InsideCircle():
PCORNER = the bounding box's 4 corners
the points PLAT on the bounding box's sides (there are either none or 2) which share the same latitude as the circle's center, if LATC is between LAT1 and LAT2, in which case these points have the latitude LATC and longitude LONG1 and LONG2.
the points PLONG on the bounding box's sides (there are either none or 2 or 4!) which share the same longitude as the circle's center. These points have EITHER longitude = LONGC OR longitude PANG(LONGC-180). If abs(PANG(LONGC-LONGM)) < LONGW/2 then LONGC is a valid longitude. If abs(PANG(LONGC-180-LONGM)) < LONGW/2 then PANG(LONGC-180) is a valid longitude. Either or both or none of these longitudes may be within the longitude interval of the bounding box. Choose points PLONG with valid longitudes, and latitudes LAT1 and LAT2.
These points PLAT and PLONG as listed above are the points on the bounding box that are "closest" to the circle (if the corners are not; I use "closest" in quotes, in the sense of lat/long distance and not great-circle distance), and cover the cases where the circle's center lies on one side of the bounding box's boundary but points on the circle "sneak across" the bounding box boundary.
If all points P in PTEST return InsideCircle(P) == +1 (all inside the circle) then the circle contains the bounding box in its entirety.
If all points P in PTEST return InsideCircle(P) == -1 (all outside the circle) then the circle is contained entirely within the bounding box.
Otherwise there is at least one point of intersection between the circle and the bounding box. Note that this does not calculate where those points are, although if you take any 2 points P1 and P2 in PTEST where InsideCircle(P1) = -InsideCircle(P2), then you could find a point of intersection (inefficiently) by bisection. (If InsideCircle(P) returns 0 then you have a point of intersection, though equality in floating-point math is generally not to be trusted.)
There's probably a more efficient way to do this but the above should work.
Use the Stereographic projection. All circles (specifically latitudes, longitudes and your circle) map to circles (or lines) in the plane. Now it's just a question about circles and lines in plane geometry (even better, all the longitues are lines through 0, and all the latitudes are circles around 0)
Yes, if the box corners contain the circle-center.
Yes, if any of the box corners are within radius of circle-center.
Yes, if the box contains the longitude of circle-center and the longitude intersection of the box-latitude closest to circle-center-latitude is within radius of circle-center.
Yes, if the box contains the latitude of circle-center and the point at radius distance from circle-center on shortest-intersection-bearing is "beyond" the closest box-longitude; where shortest-intersection-bearing is determined by finding the initial bearing from circle-center to a point at latitude zero and a longitude that is pi/2 "beyond" the closest box-longitude.
No, otherwise.
Assumptions:
You can find the initial-bearing of a minimum course from point A to point B.
You can find the distance between two points.
The first check is trivial. The second check just requires finding the four distances. The third check just requires finding the distance from circle-center to (closest-box-latitude, circle-center-longitude).
The fourth check requires finding the longitude line of the bounding box that is closest to the circle-center. Then find the center of the great circle on which that longitude line rests that is furthest from circle-center. Find the initial-bearing from circle-center to the great-circle-center. Find the point circle-radius from circle-center on that bearing. If that point is on the other side of the closest-longitude-line from circle-center, then the circle and bounding box intersect on that side.
It seems to me that there should be a flaw in this, but I haven't been able to find it.
The real problem that I can't seem to solve is to find the bounding-box that perfectly contains the circle (for circles that don't contain a pole). The bearing to the latitude min/max appears to be a function of the latitude of circle-center and circle-radius/(sphere circumference/4). Near the equator, it falls to pi/2 (east) or 3*pi/2 (west). As the center approaches the pole and the radius approaches sphere-circumference/4, the bearing approach zero (north) or pi (south).
How about this?
Find vector v that connects the center of the rectangle, point Cr, to the center of the circle. Find point i where v intersects the rectangle. If ||i-Cr|| + r > ||v|| then they intersect.
In other words, the length of the segment inside the rectangle plus the length of the segment inside the circle should be greater than the total length (of v, the center-connecting line segment).
Finding point i should be the tricky part, especially if it falls on a longitude edge, but you should be able to come up with something faster than I can.
Edit: This method can't tell if the circle is completely within the rectangle. You'd need to find the distance from its center to all four of the rectangle's edges for that.
Edit: The above is incorrect. There are some cases, as Federico Ramponi has suggested, where it does not work even in Euclidean geometry. I'll post another answer. Please unaccept this and feel free to vote down. I'll delete it shortly.
This should work for any points on earth. If you want to change it to a different size sphere just change the kEarchRadiusKms to whatever radius you want for your sphere.
This method is used to calculate the distance between to lat and lon points.
I got this distance formula from here:
http://www.codeproject.com/csharp/distancebetweenlocations.asp
public static double Calc(double Lat1, double Long1, double Lat2, double Long2)
{
double dDistance = Double.MinValue;
double dLat1InRad = Lat1 * (Math.PI / 180.0);
double dLong1InRad = Long1 * (Math.PI / 180.0);
double dLat2InRad = Lat2 * (Math.PI / 180.0);
double dLong2InRad = Long2 * (Math.PI / 180.0);
double dLongitude = dLong2InRad - dLong1InRad;
double dLatitude = dLat2InRad - dLat1InRad;
// Intermediate result a.
double a = Math.Pow(Math.Sin(dLatitude / 2.0), 2.0) +
Math.Cos(dLat1InRad) * Math.Cos(dLat2InRad) *
Math.Pow(Math.Sin(dLongitude / 2.0), 2.0);
// Intermediate result c (great circle distance in Radians).
double c = 2.0 * Math.Atan2(Math.Sqrt(a), Math.Sqrt(1.0 - a));
// Distance.
// const Double kEarthRadiusMiles = 3956.0;
const Double kEarthRadiusKms = 6376.5;
dDistance = kEarthRadiusKms * c;
return dDistance;
}
If the distance between any vertex of the rectangle is less than the distance of the radius of the circle then the circle and rectangle overlap. If the distance between the center of the circle and all of the vertices is greater than the radius of the circle and all of those distances are shorter than the width and height of the rectangle then the circle should be inside of the rectangle.
Feel free to correct my code if you can find a problem with it as I'm sure there some condition that I have not thought of.
Also I'm not sure if this works for a rectangle that spans the ends of the hemispheres as the distance equation might break down.
public string Test(double cLat,
double cLon,
double cRadius,
double rlat1,
double rlon1,
double rlat2,
double rlon2,
double rlat3,
double rlon3,
double rlat4,
double rlon4)
{
double d1 = Calc(cLat, cLon, rlat1, rlon1);
double d2 = Calc(cLat, cLon, rlat2, rlon2);
double d3 = Calc(cLat, cLon, rlat3, rlon3);
double d4 = Calc(cLat, cLon, rlat4, rlon4);
if (d1 <= cRadius ||
d2 <= cRadius ||
d3 <= cRadius ||
d4 <= cRadius)
{
return "Circle and Rectangle intersect...";
}
double width = Calc(rlat1, rlon1, rlat2, rlon2);
double height = Calc(rlat1, rlon1, rlat4, rlon4);
if (d1 >= cRadius &&
d2 >= cRadius &&
d3 >= cRadius &&
d4 >= cRadius &&
width >= d1 &&
width >= d2 &&
width >= d3 &&
width >= d4 &&
height >= d1 &&
height >= d2 &&
height >= d3 &&
height >= d4)
{
return "Circle is Inside of Rectangle!";
}
return "NO!";
}
One more try at this...
I think the solution is to test a set of points, just as Jason S has suggested, but I disagree with his selection of points, which I think is mathematically wrong.
You need to find the points on the sides of the lat/long box where the distance to the center of the circle is a local minimum or maximum. Add those points to the set of corners and then the algorithm above should be correct.
I.e, letting longitude be the x dimension and latitude be the y dimension, let each
side of the box be a parametric curve P(t) = P0 + t (P1-P0) for o <= t <= 1.0, where
P0 and P1 are two adjacent corners.
Let f(P) = f(P.x, P.y) be the distance from the center of the circle.
Then f (P0 + t (P1-P0)) is a distance function of t: g(t). Find all the points where the derivative of the distance function is zero: g'(t) == 0. (Discarding solutions outsize the domain 0 <= t <= 1.0, of course)
Unfortunately, this needs to find the zero of a transcendental expression, so there's no closed form solution. This type of equation can only solved by Newton-Raphson iteration.
OK, I realize that you wanted code, not the math. But the math is all I've got.
For the Euclidean geometry answer, see: Circle-Rectangle collision detection (intersection)
So: I have the following function, adapted from a formula found online, which takes two lat/lon coordinates and finds the distance between them in miles (along a spherical Earth):
public static double distance (double lat1, double lon1, double lat2, double lon2) {
double theta = toRadians(lon1-lon2);
lat1 = toRadians(lat1);
lon1 = toRadians(lon1);
lat2 = toRadians(lat2);
lon2 = toRadians(lon2);
double dist = sin(lat1)*sin(lat2) + cos(lat1)*cos(lat2)*cos(theta);
dist = toDegrees(acos(dist)) * 60 * 1.1515 * 1.609344 * 1000;
return dist;
}
As far as I can tell this works just fine.
What I need is a second function which, using the exact same model of the Earth's geometry, takes a single lat/lon pair [A], a heading, and a distance, and outputs a new lat/lon pair [B] such that if you started at point [A], and traveled the given distance at the given heading, you'd wind up at point [B].
This is where the fact that my geometry skills have left me entirely comes into play :)
Any help would be much appreciated!
Thanks,
-Dan
I get most of those types of formulas from The Aviation Formulary.
The formula he gives is:
Lat/lon given radial and distance
A point {lat,lon} is a distance d out on
the tc radial from point 1 if:
lat=asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
IF (cos(lat)=0)
lon=lon1 // endpoint a pole
ELSE
lon=mod(lon1-asin(sin(tc)*sin(d)/cos(lat))+pi,2*pi)-pi
ENDIF
This algorithm is limited to distances such that dlon < pi/2, i.e
those that extend around less than one
quarter of the circumference of the
earth in longitude. A completely
general, but more complicated
algorithm is necessary if greater
distances are allowed:
lat =asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
dlon=atan2(sin(tc)*sin(d)*cos(lat1),cos(d)-sin(lat1)*sin(lat))
lon=mod( lon1-dlon +pi,2*pi )-pi
Note that he's using "tc" to stand for true course (in radians clockwise from North) and the distances he gives are in radians of arc along the surface of the earth. This is explained (along with formulas to convert back and forth from nautical miles) in the first section of the Formulary. Also, check out the "Implementation Notes" and "Worked Examples" on that page.