Finding the coordinates of a point on a sphere

[lulukoko]

I have three points: A, B and C, which are all on the surface of the same sphere.
I need to find the xyz coordinates of C.
What I know:
- the radius of the sphere
- the origin of the sphere
- the xyz coordinates of A and B
- the arc distance from A to C and from B to C
- the angle between AB and BC
Any ideas?
Thanks!

 

[andrewkirk]

A good start would be to first do the analogous exercise on a plane, which has Euclidean geometry, and then see what needs to be changed to make it work on a sphere, which has non-Euclidean (elliptic) geometry.

The analogous exercise for a plane is:

I have three points: A, B and C on a plane
I need to find the Cartesian coordinates (x and y) of C.
What I know:
- the origin of the plane
- the Cartesian coordinates of A and B
- the distance from A to C (call it a) and from B to C (call it b)
- the angle between AB and BC (call it alpha)

Hint: We have two unknowns - the x and y cords of C, which we call X and Y. We can write two equations in X and Y, a, b and alpha that equate the AC distance to a and the BC distance to b.

 

[lulukoko]

I know how to do this on a plane, the trouble I am having is in, as you say, adapting the solution to a sphere.

On the plane I used simple trigonometry with the coordinates of an unknown point C(X,Y) being
X= a cos(alpha) + b sin(alpha)
Y= -a cos(alpha) + b sin (alpha)
with the coordinates of A being (a, b).

Doing this on the sphere is proving to be more difficult because it would require me to have the azimuth and the polar angles of my point C, whereas all I have is the angle between AB and BC on the surface of the sphere.

In the image attached, you can see that I know the length of AB, BC and AC. I know the angle alpha. I don't know any other angles. I know the xyz coordinates of A and B. I DO also know the xyz coordinates of point D, which is on the z-axis.

 

[andrewkirk]

On the plane I used simple trigonometry with the coordinates of an unknown point C(X,Y) being
X= a cos(alpha) + b sin(alpha)
Y= -a cos(alpha) + b sin (alpha)
with the coordinates of A being (a, b).

That doesn't look correct to me. How did you derive it?

In my setup there are seven known quantities and two unknowns. You have only used three of the known quantities, so the above cannot give a correct answer.

I also note that you have stated you are using a and b as coordinates, whereas I defined them as lengths.

I think if you first get the process for deriving equations for the unknowns and then solving them completely clear for the Euclidean case, it will be much easier to apply that to the elliptic case.

 

[lulukoko]

The equations I used were the following:
Suppose you are rotating about the origin clockwise through an angle theta. Then the point (s,t) ends up at (u,v) where
u = s cos (theta) + t sin (theta)
v = -s sin (theta) + t cos(theta)

I derived it myself from basic trigonometry functions, but here is an example that used the same reasoning as I did: http://www2.cs.uregina.ca/~anima/408/Notes/ObjectModels/Rotation.htm

 

[andrewkirk]

That is for the case of rotation around the origin by angle theta. In the OP problem the angle alpha is not at the origin. It is the angle between two line segments AB and BC, neither of which is known to go through the origin or to point towards it.

 

[arydberg]

What you really need is the equation that describes the circle and sphere.

 

[Dr Transport]

I think spherical triangles would be useful here.