Derivative — polygon properties after offsetting one point Feb 2026

Consider a given polygon.

Then offset a single point on the polygon. Let’s find the changes in certain geometric properties.

The polygon is defined by a series of $(x, y)$ coordinates $(x_i, y_i) \text{ where } i \in (0, n)$ listed in counterclockwise order.

Please know that I present these derivations/formulas more as observation and insight rather than as industrial strength computer techniques.

First, the triangle. Let’s offset the top vertex horizontally a distance, $\Delta B$ .

The area is unchanged (of course). The center of gravity, $CG_x$ , shifts by $\dfrac{\Delta B}{3}$ .

Though I have given a (belabored) explicit proof above, the same result could have been seen far easier by the formula given in Centers of Gravity of the Triangle and Lean-To Trapezoid Jan 2026.

In that post, I presented the following

$CG_x = \dfrac{A+L}{3}$

Since $L$ is a constant for a given triangle, we know that $CG_x$ must shift by $\dfrac{A}{3}$ .

In fact, moving any one vertex in any triangle a horizontal distance, ${horz\_dist}$ ,
will shift the $CG_x$ by $\dfrac{horz\_dist}{3}$ .

I note that, in a general triangle:

$CG_x = \frac{1}{3} \sum_{i=0}^{2}{x_i}$

$CG_y = \frac{1}{3} \sum_{i=0}^{2}{y_i}$

Second, the general polygon.

Let’s offset the $i^{th}$ vertex by a coordinate distance, $(\Delta x_i, \Delta y_i)$ .

The $Area$ is indeed changed.

$\Delta Area = \dfrac{1}{2} \cdot [\Delta x_i (y_{i+1} - y_{i-1}) - \Delta y_i (x_{i+1} - x_{i-1})]$