Circular plane in a sphere

[iamben]

Hi everyone,

I'm no mathematician and I've found myself a bit stuck with this problem.

I have a sphere with a radius of one centred at the origin, and two straight lines that run through the centre. I also know, in spherical coordinates, the angles phi (horizontal rotation) and theta (vertical rotation) of each of those lines. These two lines define a circular plane within the sphere and I want to calculate the angle to each of those lines from some zero reference in the circular plane. I figure the obvious references are theta = pi/2 (because I'm not going to have a case where theta = 0 or pi) and phi = 0. However, I'm a bit stumped on how to calculate my angles from one of these references.

Anyone have any ideas?

Hopefully that makes sense.

Thanks.

 

[jvicens]

Hi there,

I can definitely help you with this problem. First of all, it's great that you have a clear understanding of the problem and have identified the important references in the circular plane. To calculate the angle to each of those lines from a reference point, you can use the spherical law of cosines. This law states that in a spherical triangle, the cosine of one of the angles is equal to the sum of the cosines of the other two angles minus the product of their sines. In your case, you can use this law to calculate the angle between the two lines and the reference point. Here's an example of how to do this:

Let's say theta = pi/2 and phi = 0 are your reference points. You can use these values to calculate the angle between the two lines and the reference point. First, calculate the cosine of each angle using the spherical coordinates given. For example, the first line has a theta angle of pi/6 and a phi angle of pi/4, so the cosine of this angle is cos(pi/6) * cos(pi/4) = sqrt(3)/2 * sqrt(2)/2 = sqrt(6)/4. Similarly, you can calculate the cosine of the second line's angle.

Now, using the spherical law of cosines, you can calculate the angle between the two lines and the reference point by taking the sum of the cosines of the two angles and subtracting the product of their sines. This will give you the cosine of the angle between the two lines and the reference point. To get the actual angle, you can use the inverse cosine function (or arccosine function) on your calculator.

I hope this helps you solve your problem. Let me know if you have any further questions or need clarification on any of the steps. Good luck!