# How to Define the Intersection of a Rotated “Rectangle” in Spherical Coordinates [closed]

Can anyone tell me what is the intersection between the following two surfaces?

• The portion of a radius-1 spherical surface defined by four points: (r1, θ1, φ1), (r2, θ2, φ2), (r3, θ1, φ3), (r4, θ4, φ4)

• The portion of a radius-1 spherical surface defined by four points: (r1, θ1, φ1+x), (r2, θ2, φ2+x), (r3, θ1, φ3+x), (r4, θ4, φ4+x)

So basically I'm taking a "rectangular" (in quotes because it's really a spherical surface that has four corners like a rectangle) surface, rotating it x degrees in the direction of φ, and trying to figure out how to define the intersection between the original and final surface. I'm thinking it will either be nothing (if it rotates too far and there's no overlap), or a polygon defined by up to five points.

• Is there more to this than the obvious mathematical problem? This may put it on the wrong side of the Proposed policy on mathematical questions (See also "Are math-textbook-style problems on topic?" - spoiler alert, the answer is "no") – Rubio Dec 3 '17 at 3:01
• The reason I'm asking is because I'm working on a mobile-app programming project which requires me to know this intersection. – Anthony K. Dec 3 '17 at 3:53
• This seems like it'd have a better audience, and better topicality, at mathematics.SE. – Rubio Dec 3 '17 at 4:25
• Agreed, this fits MathsSE better. The problem seems a little underdefined though: four points won’t always uniquely define a unit sphere. For a simple example, if the points form a rectangle parallel to the x-y plane, the center of the sphere can be either above or below the points. – Bass Dec 3 '17 at 8:46
• could we migrate this instead of putting it on hold? – Quintec Dec 3 '17 at 17:58