Simple Polygons

In what follows are various notes and equations dealing with simple polygons. A simple polygon is a polygon that doesn't self-intersect and has no holes.

Area and Winding

\(A\) is the signed area:

\(\begin{aligned} A &= \sum_{i=1}^ny_i(x_{i-1} - x_{i+1})\\ &= \frac{1}{2}\left(y_1(x_n - x_2) + y_2(x_1 - x_3) + \ldots + y_n(x_{n-1} - x_1)\right) \end{aligned}\)

Notes:

• Signed area is positive iff vertices are in counterclockwise order.
• Signed area is negative iff vertices are in clockwise order.
• Signed area can't be 0.

Winding

From the vertices having the smallest x-coordinate, choose the one having the smallest y-coordinate. Call this vertex B. Call the preceding and following vertices A and C respectively. The orientation matrix and its determinant:

\(O = \begin{bmatrix} 1 & x_A & y_A \\ 1 & x_B & y_B \\ 1 & x_C & y_C \end{bmatrix}\)

\(\det(O) = (x_B - x_A)(y_C - y_A) - (x_C - x_A)(y_B - y_A)\)

Notes:

• Determinant is positive iff vertices are in counterclockwise order.
• Determinant is negative iff vertices are in clockwise order.
• Determinant can't be 0.