transform longitude latitude into meters - mapping

I need a function that maps gps positions to x/y values like this:
getXYpos(GeoPoint relativeNullPoint, GeoPoint p){
deltaLatitude=p.latitude-relativeNullPoint.latitude;
deltaLongitude=p.longitude-relativeNullPoint.longitude;
...
resultX=latitude (or west to east) distance in meters from p to relativeNullPoint
resultY=longitude (or south to north) distance in meters from p to relativeNullPoint
}
i have seen some implementations of "distance of two geoPoints" but they all just calculate the air-line distance.
i think the deltaLongitude can be transformed into meters directly but the deltaLatitude depends in the Longitude. does anyone know how this problem can be solved?

To start with, I think you have your latitude and longitude reversed. Longitude measures X, and latitude measures Y.
The latitude is easy to turn into a north-south distance. We know that 360 degrees is a full circle around the earth through the poles, and that distance is 40008000 meters. As long as you don't need to account for the errors due to the earth being not perfectly spherical, the formula is deltaLatitude * 40008000 / 360.
The tricky part is converting longitude to X, as you suspected. Since it depends on the latitude you need to decide which latitude you're going to use - you could choose the latitude of your origin, the latitude of your destination, or some arbitrary point in between. The circumference at the equator (latitude 0) is 40075160 meters. The circumference of a circle at a given latitude will be proportional to the cosine, so the formula will be deltaLongitude * 40075160 * cos(latitude) / 360.
Edit: Your comment indicates you had some trouble with the longitude formula; you might have used degrees instead of radians in the call to cos, that's a common rookie mistake. To make sure there's no ambiguity, here's working code in Python.
def asRadians(degrees):
return degrees * pi / 180
def getXYpos(relativeNullPoint, p):
""" Calculates X and Y distances in meters.
"""
deltaLatitude = p.latitude - relativeNullPoint.latitude
deltaLongitude = p.longitude - relativeNullPoint.longitude
latitudeCircumference = 40075160 * cos(asRadians(relativeNullPoint.latitude))
resultX = deltaLongitude * latitudeCircumference / 360
resultY = deltaLatitude * 40008000 / 360
return resultX, resultY
I chose to use the relativeNullPoint latitude for the X calculation. This has the benefit that if you convert multiple points with the same longitude, they'll have the same X; north-south lines will be vertical.
Edit again: I should have pointed out that this is a very simple formula and you should know its limitations. Obviously the earth is not flat, so any attempt to map it to XY coordinates will involve some compromises. The formula I derived above works best when the area you're converting is small enough to consider flat, and where the slight curvature and non-parallelism of north-south lines can be ignored. There's a whole science to map projections; if you want to see some possibilities a good place to start would be Wikipedia. This specific projection is known as the Equirectangular projection, with some added scaling.

Harvesine function is what you need.
Check it out at moveable-types There are Distance, Bearing, Midpoint and other stuff Javascript implementation working really good.
UPDATE
I've found a Java implementation of Harvesine function in another stackoverflow question

There are libraries on jstott.me.uk for PHP, Java and Javascript which do this, e.g.
var lld1 = new LatLng(40.718119, -73.995667); // New York
document.write("New York Lat/Long: " + lld1.toString() + "<br />");
var lld2 = new LatLng(51.499981, -0.125313); // London
document.write("London Lat/Long: " + lld2.toString() + "<br />");
var d = lld1.distance(lld2);
document.write("Surface Distance between New York and London: " + d + "km");

Related

SQLite query with location fields inside of region

I have a database with two columns, latitude and longitude.
I wonder if it's possible in the Transact-SQL command to retrieve only the records in a defined region of the location passed as a parameter;
For example let's say you have the following records, which is all from the same location, but we're using them for testing purposes.
Now I have the SQL query:
[self.fmQueue inDatabase:^(FMDatabase *db)
{
CLLocationCoordinate2D center = CLLocationCoordinate2DMake(latitude, longitude);
CLCircularRegion *region = [[CLCircularRegion alloc] initWithCenter:center radius:400.0f identifier:#"identifier"];
NSString* queryCommand = [NSString stringWithFormat:#"SELECT * FROM %# WHERE ?EXPRESSION_ABOUT_REGION?", readingsTable];
FMResultSet* result = [db executeQuery:queryCommand];
if (completionblock)
{
completionblock([self readingsArrayFromResultSet:result]);
}
}];
I could just retrieve records and then compare for every each one with a constructed CLLocationCoordinate2D, [region containsCoordinate:CLLocationCoordinate2D], but that feels so bad performant.
I'm looking for the most performant and appropriate solution to retrieve the desired region related location records in the T-SQL query.
Issues with the Previous Approximation
The previous answer here models the Earth as a Cartesian plane, with latitude and longitude as x and y, and then draws either a circle or a rectangle around the points---the actual shape does not matter, both methods have the same properties.
This can be fine if your constraints are really loose, and they may be here in your particular instance. For the sake of making this answer more useful to everybody I've answered the problem assuming your constraints are not quite as loose. For mobile geolocation apps in general this approximation causes problems. Maybe you've seen an app that's got it wrong. The resulting distances don't make any sense. There are several flaws with the previous approach:
Bad approximation: The scale of the Earth is big so simple approximations can be off by a scale of miles. This is bad if the user is in a car and really bad if the user is on foot. Simple tricks with degrees are not an improvement over circles or squares: latitude and longitude are not evenly distributed, and you need a full approximation like Haversine to get accurate data.
Excludes good points: Unless you severely expand the shape, your sample will exclude valid points.
Includes bad points: And the more you expand the shape to capture excluded valid points, the more you capture invalid points.
Broken sort order: Without the real calculation, you won't be able to correctly sort the output by distance. This means any ordered list you produce will be way off. Usually you want the points in order of closeness.
Two calculations: If you go back and fix it, you're doing two computations.
Single-point Haversine in C/Objective-C
The selection code is already in the other answer. This is the recomputation you referenced that you'll have to perform for each point after in Objective-C or C:
/////////////////////////////////////
//Fill these in or turn them into function arguments
float lat1=center_point.latitude;
float lon1=center_point.longitude;
float lat2=argument_point.latitude;
float lon2=argument_point.longitude;
/////////////////////////////////////
//Conversion factor, degrees to radians
float PI=3.14159265358979;
float f=PI/180;
lat1*=f; lon1*=f; lat2*=f; lon2*=f;
//Haversine Formula (from R.W. Sinnott, "Virtues of the Haversine", Sky and Telescope, vol. 68, no. 2, 1984, p. 159):
float dlon = lon2 - lon1;
float dlat = lat2 - lat1;
float a = pow((sin(dlat/2)),2) + cos(lat1) * cos(lat2) * pow((sin(dlon/2)),2);
float c = 2 * asin(MIN(1,sqrt(a)));
float R = 6378.137;//km Radius of Earth
float d = R * c;
d *= 0.621371192; //optional: km to miles [statute] conversion factor
//NSString conversion?
//if (d >= .23) return [NSString stringWithFormat:#"%0.1f m",d]; //.23m ~ 400 yds
//d *= 5280; //miles to feet conversion factor
//d /= 3; //feet to yards
//int y=(int)d;
//return [NSString stringWithFormat:#"%d yds",y];
return d;
That should be all the tools you need to complete your task as discussed.
Haversine in SQLite
I would like to show you a direct SQLite-only solution, but I was never able to get Haversine to run satisfactorily directly inside of SQLite. You don't get a square root in SQLite. You don't get a pow() function, though you can repeat an argument times itself. You don't get sin, cos, or sinh. There are extensions that add some of these features. I don't know how well-supported they are compared to the base SQLite. Even with them it's going to be too slow.
People seem to recommend updating the columns with pre-computed sines. That's fine as long as you don't need to take the sine of a difference over a whole column, in which case you're writing a new column to the table every time you need to make a new calculation, which is terrible. At any rate, I'd like to show you a comparison of how slow SQLite is on computing the Haversine, but I can't get it to compute it at all. (I think my memory of SQLite being slow on this is actually a memory of MySQL on the server being slow.)
All-points Solution in Kerf
The preceding discussion I hope is a close-to-exhaustive look at what you can do with the standard tools.
The good news is if you do it right this calculation is fast on the phone. I built a thing for you that I think solves your problem in a better way. If you are willing to use another tool, in Kerf this problem is easy. I went back and committed to the repo vectorized operations for trigonometric functions so that the calculations would be fast. My iPhone 5 will do 10,000 points in 20 milliseconds, and 100,000 points in 150 milliseconds. You can do a million points in 1.5 seconds, but at that point you'd need a throbber. Disclosure as per the rules: I built it.
From the Kerf REPL:
//ARTIFICIAL POINT GENERATION ///////////////////
n: 10**4
point_lat: 80 + rand(40.0)
point_lon: 80 + rand(40.0)
mytable: {{lats: 60 + rand(n, 60.0), lons: 60 + rand(n, 60.0)}}
lats : mytable.lats
lons : mytable.lons
/////////////////////////////////////
//COMPUTATION////////////////////////
dlon: lons - point_lon
dlat: lats - point_lat
distances_km: (6378.137 * 2) * asin(mins(1,sqrt(pow(sin(dlat/2),2) + cos(point_lat) * cos(lats) * pow(sin(dlon/2) ,2))))
//distances_miles: 0.621371192 * distances_km //km to miles [statute] conversion
//sort_order: ascend distances_km
Or via the Kerf iOS SDK. Removing the semicolon at the end of a statement will allow you to log it as JSON to the terminal.
KSKerfSDK *kerf = [KSKerfSDK new];
kerf.showTimingEnabled = YES;
//Sample Data Generation
[kerf jsonObjectFromCall:#"n: 10**4;"];
[kerf jsonObjectFromCall:#"point_lat: 80 + rand(40.0);"];
[kerf jsonObjectFromCall:#"point_lon: 80 + rand(40.0);"];
[kerf jsonObjectFromCall:#"mytable: {{lats: 60 + rand(n, 60.0), lons: 60 + rand(n, 60.0)}};"];
[kerf jsonObjectFromCall:#"lats : mytable.lats;"];
[kerf jsonObjectFromCall:#"lons : mytable.lons;"];
//Computation
[kerf jsonObjectFromCall:#"dlon: lons - point_lon;"];
[kerf jsonObjectFromCall:#"dlat: lats - point_lat;"];
NSLog(#"%#", [kerf jsonObjectFromCall:#"distances_km: (6378.137 * 2) * asin(mins(1,sqrt(pow(sin(dlat/2),2) + cos(point_lat) * cos(lats) * pow(sin(dlon/2) ,2)))); "]);
To test whether points (x,y) are inside a circle with center (cx,cy) and radius r, use the equation (x-cx)² + (y-cy)² <= r².
This does not correspond to circle because longitude and latitude values do not have the same length in the earth's surface, but it's near enough.
In SQL:
... WHERE (longitude - :lon) * (longitude - :lon) +
(latitude - :lat) * (latitude - :lat) <= :r * :r
If you use a rectangle instead, you can use simpler expressions that have a chance of being optimized with an index:
... WHERE longitude BETWEEN :XMin AND :XMax
AND latitude BETWEEN :YMin AND :YMax

Earth Latitude & Longitude to Latitude & Longitude on a 3D sphere

I have never done this sort of thing before,but what I am doing is making a 3D sphere (which has earth like texture) like shape where I want to plot locations onto it using the latitude and longitude from google maps of specific locations.Now I am wondering , my sphere radius is obviously smaller than earths radius, would this still effect the position of the latitude and longitude values to xyz given the following formula:
tx = radiusOfSphere * cos(latitude) * cos(longitude);
ty = radiusOfSphere * -sin(latitude);
tz = radiusOfSphere * cos(latitude) * sin(longitude);
Yes.
tx^2 + ty^2 + tz^2 = radius^2 & math = radius^2, so you are on your sphere.
if latitude=0, then ty = 0, so you are on a circle parallel to the equator.
if longitude=0, then tz = 0 so you are on the a meridian.
Just check that you are in xyz not xzy or zyx, and that North is x>0 and East y>0, or whatever signs are compatible with point of view of your 3D rendering soft.

converting gps coordinates to opengl word in AR

i have a list of gps coordinates (long,lat) and i have my current position (long,lat).
i found out that by subtracting the two coordinates i find the relative coordinates from my position, and that coordinates i use in my AR app to draw the pois in the opengl world.
the problem is that far-away coordinates will still be too far to "see", so i want an equation to translate everything to be close to my position, but with their original relative position.
double kGpsToOpenglCoorRatio = 1000;
- (void)convertGlobalCoordinatesToLocalCoordinates:(double)latitude x_p:(double *)x_p longitude:(double)longitude y_p:(double *)y_p {
*x_p = ((latitude - _userLocation.coordinate.latitude) * kGpsToOpenglCoorRatio);
*y_p = ((longitude - _userLocation.coordinate.longitude) * kGpsToOpenglCoorRatio);
}
i tried applying Square root in order to give them a "distance limit", but their positions got messed up relatively to their original position.
This might be because GPS uses a spherical(ish) coordinate system, and you're trying to directly map it to a cartesian coordinate system (a plane).
What you could to do is convert your GPS coordinates to a local reference plane, rather than map them directly. If you consider your own location the origin of your coordinate system, you can get the polar coordinates of the points on the ground plane relative to the origin and true north by using great circle distance (r) and bearing (theta) between your location and the remote coordinate, and then covert that to cartesian coordinates using (x,y) = (r*cos(theta), r*sin(theta)).
Better again for your case, once you have the great circle bearing, you can just foreshorten r (the distance). That will drag the points closer to you in both x and y, but they'll still be at the correct relative bearing, you'll just need to indicate this somehow.
Another approach is to scale the size of the objects you're visualizing so that they get larger with distance to compensate for perspective. This way you can just directly use the correct position and orientation.
This page has the bearing/distance algorithms: http://www.movable-type.co.uk/scripts/latlong.html
I ended up solving it using the equation of the gps coordinate intercepted with the circle i want all the pois to appear on, it works perfectly. I didn't use bearings anywhere.
here is the code if anyone interested:
- (void)convertGlobalCoordinatesToLocalCoordinates:(double)latitude x_p:(double *)x_p longitude:(double)longitude y_p:(double *)y_p {
double x = (latitude - _userLocation.coordinate.latitude) * kGpsToOpenglCoorRatio;
double y = (longitude - _userLocation.coordinate.longitude) * kGpsToOpenglCoorRatio;
y = (y == 0 ? y = 0.0001 : y);
x = (x == 0 ? x = 0.0001 : x);
double slope = x / ABS(y);
double outY = sqrt( kPoiRadius / (1+pow(slope,2)) );
double outX = slope * outY;
if (y < 0) {
outY = -1 * outY;
}
*x_p = outX;
*y_p = outY;
}

Get map position when WGS-84 lat/lon when upper left and lower right corners' lat/lon are given

Suppose I have a map, for example from openstreetmaps.org.
I know the WGS-84 lat/lon of the upper left and lower right corner of the map.
How can I find other positions on the map from given WGS-84 lat/lon coordinates?
If the map is roughly street/city level, uses a mercator projection (as openstreetmap.org seems to), and isn't too close to the poles, linear interpolation may be accurate enough. Assuming the following:
TL = lat/lon of top left corner
BR = lat/lon of bottom right corner
P = lat/lon of the point you want to locate on the map
(w,h) = width and height of the map you have (pixels?)
the origin of the map image, (0,0), is at its top-left corner
, we could interpolate the (x,y) position corresponding to P as:
x = w * (P.lon - TL.lon) / (BR.lon - TL.lon)
y = h * (P.lat - TL.lat) / (BR.lat - TL.lat)
Common gotcha's:
The lat/lon notation convention lists the latitude first and the longitude second, i.e. "vertical" before "horizontal". This is opposite to the common x,y notation of image coordinates.
Latitude values increase when going in a north-ward direction ("up"), whereas y coordinates in your map image may be increasing when doing down.
If the map covers a larger area, linear interpolation will not be as accurate for latitudes. For a map that spans one degree of latitude and is in the earth's habitable zones (e.g. the bay area), the center latitude will be off by 0.2% or so, which is likely to by less than a pixel (depending on size)
If that's precise enough for your needs, you can stop here!
The more precise math for getting from P's latitude to a pixel y position would start with the mercator math. We know that for a latitude P.lat, the Y position on a projection starting at the equator would be as follows (I'll use a capital Y as unlike the y value we're looking for, Y starts at the equator and increases towards the north):
Y = k * ln((1 + sin(P.lat)) / (1 - sin(P.lat)))
The constant k depends on the vertical scaling of the map, which we may not know. Luckily, it can be deduced observing that y(TL) - y(BR) = h. That gets us:
k = h / (ln((1 + sin(TL.lat)) / (1 - sin(TL.lat))) - ln((1 + sin(BR.lat)) / (1 - sin(BR.lat))))
(yikes! that's four levels of brackets!) With k known, we now have the formula to find out the Y position of any latitude. We just need to correct for: (1) our y value starts at TL.lat, not the equator, and (2) y grows towards the south, rather than to the north. This gets us:
Y(TL.lat) = k * ln((1 + sin(TL.lat)) / (1 - sin(TL.lat)))
Y(P.lat) = k * ln((1 + sin(P.lat )) / (1 - sin(P.lat )))
y(P.lat) = -(Y(P.lat) - Y(TL.lat))
So this gets you:
x = w * (P.lon - TL.lon) / (BR.lon - TL.lon) // like before
y = -(Y(P.lat) - Y(TL.lat)) // where Y(anything) depends just on h, TL.lat and BR.lat

How do I calculate the Azimuth (angle to north) between two WGS84 coordinates

I have got two WGS84 coordinates, latitude and longitude in degrees. These points are rather close together, e.g. only one metre apart.
Is there an easy way to calculate the azimuth of the line between these points, that is, the angle to north?
The naive approach would be to assume a Cartesian coordinate system (because these points are so close together) and just use
sin(a) = abs(L2-L1) / sqrt(sqr(L2-L1) + sqr(B2-B1))
a = azimuth
L1, L2 = longitude
B1, B2 = latitude
The error will be larger as the coordinates move away from the equator because there the distance between two longitudinal degrees becomes increasingly smaller than the one between two latitudinal degrees (which remains constant).
I found some quite complex formulas which I don't really want to implement because they seem to be overkill for points that are that close together and I don't need very high precision (two decimals are enough, one is probably fine either since there are other factors that reduce precision anyway, like the one the GPS returns).
Maybe I could just determine an approximate longitudinal correction factor depending on latitude and use somthing like this:
sin(a) = abs(L2*f-L1*f) / sqrt(sqr(L2*f-L1*f) + sqr(B2-B1))
where f is the correction factor
Any hints?
(I don't want to use any libraries for this, especially not ones that require runtime licenses. Any MPLed Delphi Source would be great.)
The formulas that you refer to in the text are to calculate the great circle distance between 2 points. Here's how I calculate the angle between points:
uses Math, ...;
...
const
cNO_ANGLE=-999;
...
function getAngleBetweenPoints(X1,Y1,X2,Y2:double):double;
var
dx,dy:double;
begin
dx := X2 - X1;
dy := Y2 - Y1;
if (dx > 0) then result := (Pi*0.5) - ArcTan(dy/dx) else
if (dx < 0) then result := (Pi*1.5) - ArcTan(dy/dx) else
if (dy > 0) then result := 0 else
if (dy < 0) then result := Pi else
result := cNO_ANGLE; // the 2 points are equal
result := RadToDeg(result);
end;
Remember to handle the situation where 2 points are equal (check if the result equals cNO_ANGLE, or modify the function to throw an exception);
This function assumes that you're on a flat surface. With the small distances that you've mentioned this is all fine, but if you're going to be calculating the heading between cities all over the world you might want to look into something that takes the shape of the earth in count;
It's best to provide this function with coordinates that are already mapped to a flat surface. You could feed WGS84 Latitude directly into Y (and lon into X) to get a rough approximation though.
Here is the C# solution. Tested for 0, 45, 90, 135, 180, 225, 270 and 315 angles.
Edit I replaced my previous ugly solution, by the C# translation of Wouter's solution:
public double GetAzimuth(LatLng destination)
{
var longitudinalDifference = destination.Lng - this.Lng;
var latitudinalDifference = destination.Lat - this.Lat;
var azimuth = (Math.PI * .5d) - Math.Atan(latitudinalDifference / longitudinalDifference);
if (longitudinalDifference > 0) return azimuth;
else if (longitudinalDifference < 0) return azimuth + Math.PI;
else if (latitudinalDifference < 0) return Math.PI;
return 0d;
}
public double GetDegreesAzimuth(LatLng destination)
{
return RadiansToDegreesConversionFactor * GetAzimuth(destination);
}
I found this link
http://williams.best.vwh.net/avform.htm
given in the answer to
Lat/Lon + Distance + Heading --> Lat/Lon
This looks promising, especially the flat earth approximation given near the end.
This would work only for small differences. Otherwise you can't just "latitudinalDifference / longitudinalDifference".
I would recommend implementing a correction factor based on the longitude. I implemented a simular routine once to return all geocoded records within x miles of a specific spot and ran into simular issues. Unfortunately I don't have the code anymore, and can't seem to recall how I got to the correction number but you are on the right track.
Has anyone tested this? It does not return the correct answers
This function assumes that you're on a flat surface. With the small distances that you've mentioned this is all fine, but if you're going to be calculating the heading between cities all over the world you might want to look into something that takes the shape of the earth in count;
Your flat earth has little to do with this. The error, as you call it, is because you are calculating an initial azimuth from a point. Unless you are heading straight to a pole, you relationship to the pole will change with distance. Regardless, the above program does not return correct results.

Resources