How to check if a point(lonc,latc) lie on a great circle running from (lona,lata) to (lonb,latb)

Question

I'm trying to produce a boolean method that will return true if a point (lonc,latc) lies on a great circle arc starting at (lona,lata) and ending at (lonb,latb)

the point of the method returning true is so if you are in a location where you should be able to see this great circel the section you can see will be shown.

the jist is that you are at (lond,latd) with a small circle at 10degrees radius and I want to work out if the great circle and small circle will intersect. There will be multiple great circles but only one small circle.

I feel the simplest approach is to check any of the longitude and latitudes on the circumference of the small circle lie on a great circle line

any help will be most appreciated

Answer 1

You have three latitude-longitude pairs (θ_a, φ_a), (θ_b, φ_b) and (θ_c, φ_c) and you want to determine whether point (θ_c, φ_c) lies on the great circle determined by (θ_a, φ_a) and (θ_b, φ_b). You can accomplish this for example using the following calculations:

1. Convert all latitude-longitude pairs to (x, y, z) triples using the following formulae: x = sin(θ)*cos(φ), y = sin(θ)*sin(φ), z = cos(θ). This will give you three triples (x_a, y_a, z_a), (x_b, y_b, z_b), (x_c, y_c, z_c).

2. Determine the formula in cartesian coordinates of the plane going through points (x_a, y_a, z_a), (x_b, y_b, z_b) and the origin (0, 0, 0). The formula is x+b*y+c*z=0 and we seek b and c which can be determined from two simultaneous equations obtained by substituting coordinates of points A and B for x, y and z in the plane formula x+b*y+c*z=0.

3. Calculate the distance between point (x_c, y_c, z_c) and the plane determined in point 2 using the following formula: d=abs(x_c+b*y_c+c*z_c)/sqrt(1+b*b+c*c).

4. From the distance in cartesian coordinates you can find the angular distance between the point (x_c, y_c, z_c) and the great circle determined by (θ_a, φ_a) and (θ_b, φ_b) using the following formula: α = asin(d).

5. Since you should not compare floating point numbers exactly you should have an angular threshold which determines how far a point can be from the great circle for you to still consider the point to lie on the circle. You then compare α determined in point 4 with the threshold to come up with the boolean value you seek.