Solving the Great Circle Problem on a Sphere

[davidam]

Homework Statement

I have a problem where I have a sphere with radius R centered at (0,0,0), and two given points on the surface of the sphere A=(a1,a2,a3) and B=(b1,b2,b3). I have to find/derive a parametric equation to describe the path from point A to point B along the surface of the sphere.

Homework Equations

I need to provide x(t)=?, y(t)=?, z(t)=? in general terms to describe the path between any two points on the surface of a sphere with any radius.

The Attempt at a Solution

When giving values to the points I can figure out how to find a plane that passes through the center of the sphere and the two points, but I don't know how to make a parametric equation to describe the circle of intersection between the sphere and the plane. If I could do that, I could set the parameters to only give me the portion of arc from A to B. I'm not exactly sure that I'm on the right track with the plane/sphere intersection idea, so any help would be much appreciated.

 

[rootX]

First, try the problem in two dimensions. You can use sin and cosine to define the circle. And then use rotations to rotate the circle to 3d. (This approach looks rather hard but not really if you can know matrices approach to do that) I think you do need to know the plane in which all three points fall.

Using above here's What I will do:

-find the plane
-find angle it makes with the z axis
-use matrices to rotate the plane to x-y plane
-solve the problem
-rotate back to the original problem using matrices again

I cannot think of any easier way at the moment.

 

[HallsofIvy]

A great circle is on a plane passing through the center of the circle and so will be the plane containing the points (0,0,0), (a1,a2,a3), and (b1,b2,b3). Find the equation of that plane: Ax+ By+ Cz= D for some numbers A,B,C,D. The equation of the sphere is x^2+ y^2+ z^2= R^2 where R^2= a1^2+ a2^2+ a3^2= b1^2+ b2^2+ b3^2 (and those last two must be equal in order that (a1, a2, a3) and (b1, b32, b3) lie on the same sphere). You have two equations in 3 variables so you can solve for two of them in terms of the third. That will give you parametric equations for the circle with that third variable as parameter.