This simple and efficient algorithm determines whether a point

is located inside a convex polygon or not. Suppose the polygon has

vertices.

First the polygon is translated by

, so that

becomes the new origin.

Next the angles of all pairs of adjacent vectors pointing from the origin to the vertices of the polygon are calculated.

If all the angles are in the range

to

(when the polygon is defined counterclockwise) or all in

to

(in the clockwise case), then the point is inside the convex polygon.

The angle calculation can be accomplished as follows:

,

where

is the rotation matrix through

.

The dot product with

reverses the

,

coordinate values and adjusts one of their signs; it is used so that only arctan and the dot product are needed.

Fortunately we only need the Boolean information of the sign of the angle, so the necessary calculation simplifies to

. This further expands to

.

Note:

must be treated mod

.

Further we can see that

; this is the area spanned by the two vectors, giving us the total signed area of the polygon:

.

So the total costs for the test are just two additions (for the initial origin translation), two multiplications, one subtraction, and one "greater than zero" comparison for every vertex; finally an

-fold equality comparison if all the signs of the angles are equal. No floating-point calculations are needed in the case of integer coordinates.

Additionally, the signed area of the polygon is calculated to show that both orientations of polygons are handled correctly.

This Demonstration was developed before fully understanding the

Point in Triangle Demonstration. The slight difference is the use of the DotProduct (and perhaps a better documentation/explanation and a more compact implementation) in contrast to the Determinant used in

Point in Triangle.

Similar algorithms exist which unnecessarily sum up all the mentioned angles (not only their signs), at the relative high costs of arctan and floating-point operations.