A point on a great circle

[..]

Given longitude and latitude of 2 points on a shpere, there is an unique great circle passing through both points. Is there any formula to get the latitudfe/longitude of any point on this great circle?

At first I guess that, given 2 points AB on a great circle, any point C on the great circle can be found by:
latitude_C = latitude_A + (latitude_B - latitude_A) / angle_BOA * angleCOA;
(longitude similar)

However, I find that there is something wrong on the value I found.
So I think my guess is wrong......Do anyone know about that?
Thanks in advance.

[ericschmidt]

Hi!

Well that formula is not correct - it looks like a linear interpolation - but that does not work on sphere. One counterexample would be: latidude_A = latitude_B = 45 deg ... then latitude_C would also have to 45 deg (for all C-points) - but that is not a great circle.

To get the unique great circle through the two points you need the vector n perpendicular to this great circle which is in the plane defined by O, A and B. You can compute this vector with the "x-product" of the vectors a and b:

n = a x b

To get the equation for points (c) on the great circle you can cunstruct a new vector m which is perpendicular to a and n and is therefore in the plain of the great circle:

m = n x a

now m needs to be trimmed to the right length (the radius of the sphere r = |a|):

m_r = m * (r / |m|)

all points c may have an arbitrary angle "angle_COA" in the range of [0, 360] deg and can be calculated from:

c = a * cos(angle_COA) + m_r * sin(angle_COA)

note that "*" indicates the multiplication of a vector with a scalar

You can interpret a and m_r as the new coordinate axes in 3 dimensional space for a normal two axes circle.

Of course the according formulas with b would also work.

I hope that helps.

Yours, Eric