How to calculate a solid angle defined by a point and a polygon?

Question

Let's define a point O with coordinates Xo, Yo, Zo and a polygon ABCD with coordinates (Xa,Ya,Za), (Xb,Yb,Zb), (Xc,Yc,Zc), (Xd,Yd,Z)

What is the most usual algorithmic way to compute the solid angle Ω defined by the polyhedron ABCDO?

Thank you

Answer 1

Rather simple method is to divide this pyramid into two tetrahedrons (for example, with common diagonal AC, if ABCD is convex), then calculate solid angle for each tetrahedron

Ω  = 2 * ArcTan(Dot(u1 X u2, u3) /(1 + Dot(u2,u3) + Dot(u1,u2) + Dot(u1,u3)))

where u1, u2, u3 - normalized (unit) vectors from O to A, B, C point (for 1st tetrahedron)

Look at the possible issues of this approach in Wiki