About points in the circle of sphere surface

Question

here is the image

Hey guys! I’m so sorry for the silly question but my math skills are very poor and I just need this problem fixed. I made a simple image about it and I hope it won’t confused you. In the image you can see 1. There is a Sphere with center point O 2. A cone which vertex is at O and radius is same as the sphere. The cone divides the sphere surface into two parts (red and blue) 3. the angle of the cone is α 4. Ray OC passes through the center of the base circle of the cone and intersects sphere surface at point C, which sphere coordinate is (θc, φc) 5. Point A (θa, φa) is a point on the sphere surface

The question is how to determine whether point A is in the red part of sphere surface according to its sphere coordinate(θa, φa)? Thank you very much!

Answer 1

If $r$ is a radius of sphere, then in spherical coordinates points $C$ and $A$ have coordinates $(r,\theta_c,\varphi_c)$ and $(r,\theta_a,\varphi_a)$.

Then using help from this question, you can find the angle between $\vec{OC}$ and $\vec{OA}$ and compare it to $\alpha/2$.

Answer 2

$A$ is in the red region if \begin{equation} \vec{OA}\cdot \vec{OC}> R^2 \cos(\frac{\alpha}{2}) \end{equation}