NAN value and computing angle in equatorial plane with great
I am developping a small code with which I need to compute the difference of angles in equatorial plane (ie the difference of longitudes) as a function of the angles difference in a great circle plane (parameterized by a given latitude).
I used the following formula from this wikipedia link :
d(sigma) = arcos (sin(phi1).sin(phi2) + cos(phi1).cos(phi2).cos(d(lambda))
The goal is to compute d(lambda) difference of angle. In my code, input parameters are :
radius = 50 phi1 = 0 phi2 = initial latitude describe below d(sigma) = (distance / theta) where theta is the local angle in great circle plane and distance is the perimeter of this great circle.
local angle theta in great circle plane starts from 0 and is incremented by 0.01 step .
Knowing phi1 , phi2 , distance and theta , I can express d(lambda) as (in Javascript language) :
var distance = radius*Math.abs(theta);
var deltaLambda = Math.acos(Math.cos(distance/radius) / Math.cos(angleTheta));
where angleTheta is the latitude of starting point (identified by coordTorus THREE.Vector3) and equal to :
var angleTheta = Math.atan(coordTorus.y / Math.sqrt(coordTorus.x * coordTorus.x + coordTorus.z * coordTorus.z));
My issue is that for an initial value of angleTheta equal to 0 , an initial theta value equal to 0 , then the computing of deltaLambda is good but not in other cases :
Let's take for example an initial value of angleTheta = PI/4 and theta = 0 , then I have an NAN value for deltaLambda because in the above formula, I get :
var deltaLambda = Math.acos(Math.cos(0.5/50) / Math.cos(Math.PI/4));
So I get Math.acos(sqrt(2)) = NAN
How could I circumvent this issue and find a trick with which the value inside Math.accos remains into [-1,1] interval ?
I saw on above link there were others formulas for computing great-circle distance but I need to isolate the d(lambda) variable with these formulas, I mean a symbolic expression of d(lambda) as a function of others parameters.
If someone could give another consistent formula or find a way to avoid NAN value error , this would be nice.
Thanks in advance.
It impossible that for lat1=0, lat2=45 great circle distance is 1/100 of radius! Minimal possible d is Sqrt(2)/2 * R . So you take illegal starting data for calculations.
Another issue - wrong formula to get latitude from Cartesian coordinates. Right one:
lat = Arccos(z/R)
or
lat = atan(Sqrt(x^2+y^2) / z)
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