CODEWITHSUNDEEP
javascriptgoogle-mapsmath2dcoordinates
Антон Онищенко
Антон Онищенко

Reputation: 137

How to calculate northEast coordinate by radius and center on map

I have this formula: 

var bounds = map.getBounds();

var center = bounds.getCenter();
var ne = bounds.getNorthEast();

// r = radius of the earth in statute miles
var r = 3963.0;  

// Convert lat or lng from decimal degrees into radians (divide by 57.2958)
var lat1 = center.lat() / 57.2958; 
var lon1 = center.lng() / 57.2958;
var lat2 = ne.lat() / 57.2958;
var lon2 = ne.lng() / 57.2958;

// distance = circle radius from center to Northeast corner of bounds
var dis = r * Math.acos(Math.sin(lat1) * Math.sin(lat2) + 
  Math.cos(lat1) * Math.cos(lat2) * Math.cos(lon2 - lon1));

It calculates radius by NE and center. I need: Formula to calculate NE coordinate by given radius and center on map written with js.

Upvotes: 0

Views: 890

Answers (1)

mikuszefski
mikuszefski

Reputation: 4043

Ok, first I guess it will help to understand where the used formula comes from. And even before that, note that I will use standard mathematical coordinates. This differs from geographical long/lat but should be easy to transform

Hence a point on a sphere is (x,y,z)= r*(cos p sin t, sin p sin t, cos t). So p is the angle from x to y and t is the angle of the z-axis.

If you have two points (p,t) and (q, u) we can rotate the first point to p=0, i.e. over the x-axis. Than the points have coordinates (0,t) and (q-p,u). Now we rotate the points around y such that the first point becomes the north pole.

[ cos t, 0, -sin t]   [x]     [ cos t, 0, -sin t]   [ cos(q-p) sin(u)]
[    0   1,   0   ] . [y]  =  [    0   1,   0   ] . [ sin(q-p) sin(u)] 
[ sin t, 0,  cos t]   [z]     [ sin t, 0,  cos t]   [       cos(u)    ]

the new z than is

z_new = sin(t) cos(q-p) sin(u) + cos(t)cos(u)

Naturally the arc length from here to the north pole is just 

alpha = arcsin( sin(t) cos(q-p) sin(u) + cos(t)cos(u) )

and for the true distance we have to multiply with the radius r of the sphere.

Now the other way. we have a point (p,t) and want the (q,u) given that its direction is an angle beta off north and in distance d. In a fist step we set point (p,t) as north pole. That makes the second point (Pi + beta, d/r) (Note angles are mathematical positive if ccw). This system has to be rotated such that the north pole goes to the given (p,t). This is done by

[  cos t, sin t,  0]   [ cos p, 0,  sin p]   [x]   
[ -sin p, cos t,  0] . [    0   1,   0   ] . [y]  
[   0   ,  0   ,  1]   [ -sin p, 0, cos p]   [z]

setting (Pi + beta, d/r) = (gamma, theta) we get

z_new = -sin(p)cos(gamma)sin(theta)+cos(p)cos(theta)

consequently:

u = arccos( z_new )

Finally: 

x_new = cos(t) ( cos(p)cos(gamma)sin(theta) + sin(p)cos(theta) ) + sin(theta)sin(gamma)sin(theta)

As x_new = cos(q)sin(u) and we know u

q = arccos( xnew / sin(u) ) = arccos( xnew / sqrt( 1 - z_new ) )

I hope I got it all right and remember this is in typical mathematical polar coordinates and it has to be translated to the sin/cos usage and angle definition in geography.

Upvotes: 1

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