Can there be a single equation for more than one equilateral polygon with the origin set at zero?

Question

I have been trying to make a single equation for more than one polygon by just changing the numbers that the variables represent and I can't figure out how to do so, so any help with that would be nice.

Answer 1

There exists a general implicit equation, valid for any regular polygon (same sidelengths, same angles), centered in the origin with $n$ vertices:

$$f(x,y)=\sum_{k=1}^n \ \underbrace{|x \cos(r+ka)+y \sin(r+ka)-p|}_{d_k}=np \tag{1}$$

where

• $a:=\frac{2\pi}{n}$

• $r$ is an angular value giving the rotation angle of the figure around the origin.

• $p$ gives the overall size of the polygon.

Here is a particular case with $n=3, p=3.5, r=-1.2$:

Proof: Expression (1) is due to a generalization of Viviani theorem expressing the fact that the sum of distances $d_k$ from any interior point of a regular polygon to its sides is a certain constant. See for example here.

It is interesting to consider the surface with equation

$$z=f(x,y)=\sum_{k=1}^n \ |x \cos(r+ka)+y \sin(r+ka)-p|\tag{2}$$

Please note the difference between (1) and (2): the LHS is no longer restricted to value $np$.

The structure of this surface shows that $z=np$ is a limit case corresponding to a filled triangle, and as soon as we are on a "level set" $np+\varepsilon$, we have a triangle reduced to its sides ; a triangle ? Not really. A triangle with "chamfered" vertices... a phenomena that takes more and more importance as $z$ increases...

Remark; There are rather similar formulas for other types of shapes, for example this one gives a quarter plane that can take any position in the plane:

$$f(x,y)=|x \cos(a+c)+y \sin(a+c)|+x \cos(2a+c)+y \sin(2a+c)=b$$

See as well this reference.