# Area moments of a Freeman chain contour

The area moments of a polygon can be computed by generalizations of the shoelace formula for area.

In particular, the first and second order moments are given by https://en.wikipedia.org/wiki/Polygon#Centroid and https://en.wikipedia.org/wiki/Second_moment_of_area#Any_polygon.

I was wondering if there are simplified forms when the polygon is described by a Freeman chain (so that the coordinates vary by at most one unit from vertex to vertex).

I would solve the problem as follows. First, I need an analytic expression for an arbitrary polygon. Say that you have $$n$$ elements of length $$S_k$$ and angles $$\Phi_k$$ Then you can express the polygon in the complex plane with the fractal linkage (Gnomon: from Pharoahs to fractals, Michael J. Gazal$$\acute{\text{e}}$$, Princeton University Press, 1999) , that is

$$z=\text{cum}\sum_{k=1}^n S_ke^{i\Phi_k}$$

The perimeter, area, and centroid are given by

$$s=\int |\dot z| du\\ A = {\textstyle{1 \over 2}}\int {\Im\left\{ {{z^{_*}}\dot z} \right\}} \,du\\\ z\left( G \right) = \frac{{\int {z\Im\left\{ {{z^{_*}}\dot z} \right\}} \,du}}{{3A}}\$$

Zwikker, C. (1968) [The Advanced Geometry of Plane Curves and Their Applications, Dover Press] gives the derivations for the following moments:

$$\text{Polar inertia}\\ J_0=\frac{1}{4}\int zz^{_*} {\Im\left\{ {{z^{_*}}\dot z} \right\}} du\\ \text{Binet's inertia moments}\\ J_x=\frac{1}{4}\int x^2 {\Im\left\{ {{z^{_*}}\dot z} \right\}} du\\ J_y=\frac{1}{4}\int y^2 {\Im\left\{ {{z^{_*}}\dot z} \right\}} du\\ \text{Centrifigal inertia moment}\\ J_{xy}=\frac{1}{4}\int xy {\Im\left\{ {{z^{_*}}\dot z} \right\}} du\\$$

Notice that $$J_0=J_x+J_y$$.

At least, that's what I would do.

• Thank you for this input, Cye. I do have the formulas for a polygon (see the links). What I am after is the specialization for the case of a Freeman chain. In fact, a micro-optimization. – Yves Daoust Jun 29 at 8:01