Green's formula to calculate the area of a polygon

Introduction to Algorithms Chapter 31, Section 1, Question 8. As long as it is a simple polygon whose edges do not intersect, that is, not only convex polygons, but also various odd-shaped concave polygons, the area can be calculated using Green's formula.
Green's formula: If the functions P(x,y), Q(x,y) are continuous on the closed region D surrounded by one or more smooth curves, and have continuous first-order partial derivatives, then there is
$\iint_d(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})d\sigma = \oint_LPdx + Qdy$
L as the region The boundary curve of D, and take the positive direction.
A simple polygon with disjoint edges is just a closed area enclosed by several line segments, so Green's formula can be used.
Let P=0, Q=x, then the area S = $\int_Dd \ sigma = \ oint_Lxdy$
Let the coordinates of the i-th point be , and $P_i$ the coordinates of $(x_i, y_i)$ the i + 1-th point $P_{i + 1}$ be , then the parameter formula of $(x_{i + 1}, y_{i + 1})$ the line segment is , so , so the area . The above formula can calculate the area of any simple polygon, including triangles, quadrilaterals, and hexagons. $\overline{P_iP_{i + 1}}$ $\begin{cases}x = x_i + (x_{i + 1} - x_i)t \\y = y_i + (y_{i + 1} - y_i)t\end{cases}$
$\int_{\overline{P_iP_{i + 1}}}xdy = \int_0^1(x_i + (x_{i + 1} - x_i)t)(y_{i + 1} - y_i)dt=\frac{1}{2}(x_i + x_{i + 1})(y_{i + 1} - y_i)$
$S=\frac{1}{2}|\sum_{i=1}^{n}(x_i + x_{i + 1})(y_{i + 1} - y_i)|$

python implementation:

# P is list of vertices of the polygon
def polygon_area(P):
    n = len(P)
    P.append(P[0])
    S = 0
    for i in range(0, n):
        S = S + (P[i][0] + P[i + 1][0]) * (P[i + 1][1] - P[i][1])
    return 0.5 * abs(S)

Author: Wang Er
Link : https://www.zhihu.com/question/53259589/answer/134574326
Source: Zhihu The
copyright belongs to the author. For commercial reprints, please contact the author for authorization, and for non-commercial reprints, please indicate the source.

Green's formula to calculate the area of a polygon

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