# Degrees of freedom of a 2D shape

Let the length of sides and internal angles be the $2n$ paramaters defining a n-gon. How many of them are needed to define it completely without over specifying?

In "A treatise on Algebra" George Peacock stated that a polygon is defined by three equations. Let $a_i$ (with $i=1,2,..,n$) be the lengths of the sides; and $\theta_{i,i+1}$ the internal angles between them. Then:
$$a_1+\sum_{i=2}^{n}(-1)^{i-1}a_i\cos\left[\sum_{j=2}^i\theta_{j-1,j}\right]=0\\ \sum_{i=2}^{n}(-1)^ia_i\sin\left[\sum_{j=2}^i\theta_{j-1,j}\right]=0$$
are known as the "equations of figure". While $$\sum_{i=1}^n\theta_{i,i+1}=(n-2)\pi$$ is the "equation of angles".
These equations make up for not knowing up to three angles. Therefore a polygon is completely determined by $2n-3$ parameters. However they can only be split in three situations: $n$ sides and $n-3$ angles, $n-1$ sides and $n-2$ angles, and $n-2$ sides and $n-1$ angles.
• So the degrees of freedom of an n-gon is $2n - 3$. Oct 16, 2018 at 8:48