Context: I apologize in advance of the non-rigorousness of the question, as it turned out to be harder to formulate than I originally thought. I'm going over different ways to find two "bounding" vertices in 2D space of a given polygon w.r.t. to a known point. In this context by "bounding" vertex I mean the situation best described by this image. I.e. let p be the known point and imagine that we place a light source at p. Then the bounding vertices of a polygon P(x_1,...,x_n) are those two points v_1, v_2 for which the connected line segment l(v_1, v_2)blocks the light from pin the same way as the whole polygon P(x_1,...,x_n) does.
Question: I already have a solution which compares vertices of P by the rotation angle w.r.t. to p. However this method requires using the trigonometric atan2 function, so I'm interested in knowing is there a computationally cheaper method.