A Visual Guide to Archimedean Spirals and its Special Cases

Trace of an Archimedean Spiral

The Archimedean Spiral is a curve that is one of several types of spiral functions. It is traced out by a curve which moves outwards from a point, moving further away as it revolves around that point at a constant speed. A classic example of this spiral in action in real life can be seen in vinyl players.

There are many aspects of the Archimedean Spiral to explore and this article will cover a few of these in a condensed fashion. Normally spirals are described using polar coordinate systems however I have decided to present them in their cartesian forms which I found to help my intuition in understanding them and easy to explore their behaviour as well.

The equation of the Archimedean Spiral in its cartesian form is described by the parametric equation below and has parameters k and p.

Branches

Let's first look at the two branches of the Archimedean Spiral which represent the two cases for the equation where t > 0 and t < 0:

A plot of the Archimedean Spiral from eq 1, where k = p = 1.5 where blue is the branch t > 0 and red is the branch t < 0.

The branch of the spiral for t > 0 is anti-clockwise and the branch of the spiral for t < 0 is clockwise. You can see that if one was flipped along the y…