Finding the surface of a plane that is projected on a sphere

Question

I would like to find the surface of a plane that is being projected on a sphere (as shown in the picture). I've tried to find online how to do this projcetion, but I have not been able to find it. For my assignment I need to find the surface of a football goal with the center being the penalty mark. A player can shoot the ball in any direction (, but not -z, so half sphere). Thus, I have to find the ratio between surface of the football goal and half of a sphere. The distance between the penalty mark and the center of the goal is 11 yards, height of the goal is 8 foot, and the width is 24 foot. Is there a general formula for how to do this? Or does anyone know in what direction I should look?

Plane projection on a sphere

Answer 1

Attach a coordinate frame to the sphere with its origin at the center of the sphere.

Express the four corners of the goal in that coordinate frame as $ P_i = ( x_i, y_i, z_i) , i= 1, 2, 3, 4$. Generate the parametric equation of the border of the goal as the $4$ line segments connecting $P_i$ to $P_{i+1} $ (with $P_5 = P_1$). This equation is

$Q_i(t) = (1 - t) P_i + t ( P_{i+1} - P_i ) $

Now project $Q_i(t)$ on the sphere, the formula is very simple

$S_i(t) = R \dfrac{ Q_i(t) } { | Q_i(t) | } $

where $R$ is the radius of the sphere, and $|Q_i(t)|$ is the length of the vector $Q_i(t)$.

Hence, the projection of the goal is the concatenation of $\{ S_i \}, i = 1, 2,3,4 $