Is a point inside regular hexagon

Question

I'm looking for advice on the best way to proceed. I'm trying to find whether a given point A:(a, b) is inside a regular hexagon, defined with center O:(x, y) and diameter of circumscribing circle.

It seems like overkill to use Ray-casting, or Winding-number to determine this, for such a simple case, and I'm currently looking at the option of finding the angle (from horizontal) of the line OA, and "normalising" (probably not the right word) it into one of the 6 equilateral triangles and seeing if this new point lies within this triangle.

I get the feeling I'm missing something simple, and there's an easy way (or if I'm really lucky, a Java API) to do this simply and efficiently.

Thanks for your help.

Edit: The hexagon is oriented such that one of the sides is flat with the horizontal.

Answer 1

I would first check if the point is inside the inscribed circle (you can compute the inscribed circle radius easily) or outside the circumscribed circle (that you already have).

The first means the point is in, the latter means it's out.

Statistically, most of the input points should allow you to decide based on the above simple tests.

For the worst case scenario (point is in between the inscribed and circumscribed circles), I think you can find the two vertices that are closest to the point and then see on which side of the segment V1V2 the point is (inner or outer, as relative to the O center). Special case: point is equal to one of the vertices => it's in.

If I'll have a more clever idea (or if I'll ever start to really learn trigonometry), I'll edit the answer to let you know :)