User:Sdikiy/sandbox

Centroid Formula


 * $$(2) \ x_i = X + \delta x_i ;\ y_i = Y + \delta y_i\;$$


 * $$(3) \ A = \frac{1}{2}\sum_{i=0}^{n-1} ((X + \delta x_i)\ (Y + \delta y_{i+1}) - (X + \delta x_{i+1})\ (Y + \delta y_i))\;$$


 * $$(4) \ A = \frac{1}{2}\sum_{i=0}^{n-1} ((X Y + X \delta y_{i+1} + Y \delta x_i + \delta x_i \delta y_{i+1}) - (X Y + X \delta y_i + Y \delta x_{i+1} + \delta x_{i+1} \delta y_i))\;$$


 * $$(4^o) \ A = \frac{1}{2}\sum_{i=0}^{n-1} ((X Y + X \delta y_{i+1} + Y \delta x_i + \cancel{\delta x_i \delta y_{i+1}}) - (X Y + X \delta y_i + Y \delta x_{i+1} + \cancel{\delta x_{i+1} \delta y_i}))\;$$


 * $$(5) \ A = \frac{1}{2}\sum_{i=0}^{n-1} ((X Y + X \delta y_{i+1} + Y \delta x_i) - (X Y + X \delta y_i + Y \delta x_{i+1}))\;$$


 * $$(6) \ A = \frac{1}{2} (X (\sum_{i=0}^{n-1} \delta y_{i+1} - \sum_{i=0}^{n-1} \delta y_i) + Y (\sum_{i=0}^{n-1} \delta x_i - \sum_{i=0}^{n-1} \delta x_{i+1}))\;$$


 * $$(7) \ A = \frac{1}{2} (X (\delta y_n - \delta y_0) + Y (\delta x_0 - \delta x_n)) = 0\;$$