Consider an arbitrary 2D polygon. I define the unit normal of a side of the polygon the vector orthogonal to the side of the polygon and pointing outward, of length one.
I want to know what is the family of polygons such that the vector sum of all unit normals of the polygon amount to zero.
For example, we know that a rectangle would be in this family, because the opposite normals cancel out. However, an isosceles trapezoid would not fulfill this rule, because the sum would be positive in the direction of the smaller base.
What is this family? Are there any interesting properties in this family? I do not require that the polygon should be convex.
Convex polygon set
$n$ number of of regularly oriented / spaced in ($\theta= \pi/n ) $ interval vectors $F$ add up to zero.
The zero sum is seen trigonomically either using formula ang $y$ ox $x$ direction.
$$ \sin (1 \pi/n )+\sin (2 \pi/n ) + ... + \sin ( n\cdot2 \pi/n )=0 $$
$$ \cos (1 \pi/n )+\cos (2 \pi/n ) + ... + \cos ( n\cdot2 \pi/n )=0 $$
or
graphically in a vector diagram with vectors connecting end to end along variable directions tangent to a circle of radius $F$. For example $n=8$
The vectors can be placed either at end of each vertex or at middle point of each side of regular polygon, the $n-$gon
$$ \Sigma_1^n F \cos \theta_i =0 ;\, \Sigma_1^n F \sin \theta_j =0 $$
The above is a statement of static equilibrium of rotating forces $F_{i}$. $F$ can have any value including unity.
In Three dimensional force equilibrium we can assign directions to each directed area of a Platonic solid which are further amenable to trigonometric generalization by integrals/sums to zero now in three mutually perpendicular directions.
Non-convex polygons set
A non convex set is obtained by superposition of two or more convex sets.
we have for $x$ or $y$ vector balance in each direction
$$ \Sigma_1^{n} F_i \sin \theta_i =0 , \Sigma_1^{n} F_k \sin \theta_k =0 , $$
where $k$ is a multiple of $i$.