determine if line segment is inside polygon

Question

suppose we have convex polygon with vertices

(v0,v1,....vn)

my aim is to determine if for given point p(x,y) any line segment

Answer 1

Assume a point P and a convex polygon with n vertices V_1 to V_n (n > 2).

Sort the vertices of the polygon by their angle relative to a selected vertex, so that they are in clock-wise or counter-clockwise order. The edges of the polygon are then V_1 -> V_2, V_2 -> V_3, ..., V_(n-1) -> V_n, V_n -> V_1.

Now, for every edge, check the value of the cross product (V_(i+1) - V_i) x (P - V_i). Now P is inside the polygon iif all the values are >= 0 or all the values are <= 0.

There's a good tutorial on TopCoder for the more general problem where the polygon doesn't have to be convex. What they do is send a ray from the test point and check how many edges it intersects.

NOTE: The cross product used here is defined as (u1, u2) x (v1, v2) := u1*v2 - u2*v1