Locus of segments of length $n$ which form an $n^\circ$ angle with the $x$-axis

Question

So basically, I was wondering what would happen if you take the locus of line segments of length $n$ that make a $n^\circ$ angle with respect to the $x$ axis. I did this, and this is what I got:

I was wondering if there is any properties of this shape and if it is well known in any field of study?

Answer 1

Let’s consider the graph in polar form: $$r=\frac{180}{\pi}\theta$$ Technically, this is actually the locus of all of the endpoints of the line segments in your diagram. Here’s another picture:

This shape is called an archimedian spiral. Not sure what kind of properties you find “interesting,” but here are a few:

• The area of the region shaded by line segments in your picture is equal to $15\pi/4$.
• The perimeter of the region in your picture is approximately equal to $209.13$, but its exact length is given by $$90+\frac{45\sqrt{4+\pi^2}}{2}+\frac{90\sinh^{-1}(\pi/2)}{\pi}$$
• The graph of this spiral in cartesian coordinates is given by $$\frac{\pi x}{180}\tan\Big(\sqrt{x^2+y^2}\Big)=y$$

Some interesting applications can be found here.