how to calculate geometric distortion value after transformation

Question

I am applying different transformations to various polygons and want to find geometric distortion based on area and perimeter for each transformation. I am calculating area and perimeter distortions seperately and then want to integrate these values to find an average distortion(see the figure). The main aim is to find the transformation with the least distortion. Do you have any suggestion to integrate these values in an optimum way?

examples

Answer 1

Not really an answer to your somewhat vague question, but some ideas that may help.

Fortunately, you seem to be using relative rather than absolute change. You can try the arithmetic average of the relative changes in area and perimeter as your "average distortion". For a little more flexibility, you can try a weighted average of those two. The figure with "least distortion" will depend on how you choose the weights.

I suggest you do the algebra to experiment converting a rectangle with sides $L$ and $W$ to a square of side $S$ in several ways to see what happens to the relative changes $\Delta P = 4S/(2L + 2W)$ and $\Delta A = S^2/LW$.

Edit in response to comment.

Summing them is probably a bad idea, since the range for each is $0$ to $\infty$ with $1$ meaning "no change". So my suggestion above that you average them is wrong. You should consider the geometric mean: $$ \sqrt{\Delta A \Delta P} $$ or just the product $\Delta A \Delta P$ or a weighted geometric average $$ (\Delta A)^a(\Delta P)^p $$ where you choose the exponents to make the result look good to you.

The further this quantity is from $1$ the more distorted your figure.

Whether or not any of this is "sensible" isn't a mathematical question. It is if it does what you need your average distortion measure to do. Note that a geometric mean of $1$, which would come from no change at all might come from a smaller perimeter and a larger area.