Parametrization of a regular planar polygon with an arbitrary number of sides

[epr2008]

I was wondering if anyone knew of a common technique for parametrizing a regular polygon with an arbitrary number of sides. I figured such a problem would be easy or at least be well documented online, but that doesn't seem to be the case.

I started by assuming that the polygon was centered at the origin of the Polar Plane and the sides were of length R. Then since the polygon has n vertices, we can draw n line segments, each starting at the center and protruding to a vertex. Now, we know that the sides are all equal, as well as the interior angles, and that the center is equidistant from each vertex, then the angles between the circumradii are multiples of 2pi/n. These contraints also require that the circumradii bisect the interior angles, and therefore they partition the polygon into n isosceles triangles. Then the magnitude of the interior angle situated at the ith vertex is then given by

$$\left| {\angle {V_i}\left( {{P_n}} \right)} \right| = \pi - \frac{{2\pi }}{n}$$

we now use the law of sines to determine the length of the ith radius.

$$\left| {{R_i}\left( {{P_n}} \right)} \right| = \frac{{\sin \left( {\frac{\pi }{2} - \frac{\pi }{n}} \right)R}}{{\sin \left( {\frac{{2\pi }}{n}} \right)}} = \frac{R}{{2\sin \left( {\frac{\pi }{n}} \right)}}$$


Next, I transformed these pairs into their Cartesian representation because parametrizing straight line segments in polar form seemed a little inefficient. Which is simple enough but then, in attempting to construct a unit vector, computation became cumbersome very quickly. I assumed that there was a more compact form since the magnitude must be independent of i, but after attempting to use the multiple angle formulae I gave up.

So, I went back to polar form and realized that the the radius vector oscillates back and forth between the circumradius and inradius as the polar angle varies, but I can't seem to put that statement into a parametrization of the polygon.

Anyone have any suggestions? I'm thinking this has to be something simple, and hopefully is somewhat elegant. My brain is just still in full reboot mode from midterms. :(

 

[epr2008]

Btw sorry if you don't like my notation, I tend to make it up as I go along and I'm extremely OCD when it comes to notation, neglecting brevity for the sake of organization.

 

[Vargo]

Could you give a precise definition of what you mean by a parametrization of the polygon? Are you referring to the interior region or the boundary? If you are referring to the boundary "curve", then you could do something like this:

For 0<t<n, define the curve piecewise by

[(exp(i2pi/n)-1)(t-k)+1]exp(i2pi*k/n)

over the subinterval k<t<k+1.

 

[epr2008]

My bad, I meant the boundary. And I was looking for a more of a vector representation

 

[Vargo]

If you use Euler's identity and multiplication of complex numbers, then the the expression I wrote works as a constant speed, vector valued function of t that starts at (1,0) and goes counterclockwise around the polygon.

http://en.wikipedia.org/wiki/Euler's_formula

This assumes that one of the polygon's vertices is (1,0). Then each successive vertex will be given by exp(i2pi*k/n).

For example, the next vertex would be (cos(2pi/n), sin(2pi/n)). So the parametrization of the first side would be:
(1,0) + t*( cos(2pi/n) - 1, sin(2pi/n)) for 0 < t < 1.

 

[epr2008]

Yes, I know I could take the real and imaginary parts of that as the x and y vectors but I meant in terms of x and y or r and theta without introducing a parameter t. I guess it's easier for me to see what's going on this way

For instance, for n= 4

$$\begin{array}{l} {{\vec P}_4}\left( {x,y} \right) = \left( {x\hat x - R\hat y} \right) - \left( { - R\hat x - R\hat y} \right) = \left( {x + R} \right)\hat x\\ x \in \left[ { - R,R} \right]\\ {{\vec P}_4}\left( {x,y} \right) = \left( {R\hat x + y\hat y} \right) - \left( {R\hat x - R\hat y} \right) = \left( {y + R} \right)\hat y\\ y \in \left[ { - R,R} \right]\\ {{\vec P}_4}\left( {x,y} \right) = \left( { - x\left( { - \hat x} \right) + R\hat y} \right) - \left( {R\hat x + R\hat y} \right) = \left( { - x + R} \right)\left( { - \hat x} \right)\\ x \in \left[ {R, - R} \right]\\ {{\vec P}_4}\left( {x,y} \right) = \left( {R( - \hat x) - y( - \hat y)} \right) - \left( {R( - \hat x) - R( - \hat y)} \right) = \left( { - y - R} \right)( - \hat y)\\ y \in \left[ {R, - R} \right] \end{array}$$

I'm a weirdo but Idk like I said it's just easier for me to see what's going on this way. I was looking for kind of an analogous parametrization for an arbitrary value of n.

 

[epr2008]

Oh wait you're talking roots of unity! Duh! Wow my bad lol