Evolutes

An Evolute of a Circular Helix

In a previous post, we studied the involute of a curve. Now we will reverse the direction of the problem: Given a regular curve $\alpha$, another curve $\beta$ is called an evolute of $\alpha$ when $\alpha$ is an involute of $\beta$.

This means that

1. $\beta(s) - \alpha(s)$ lies along the tangent to $\beta$ at $\beta(s)$.
2. $\beta'(s)$ is orthogonal to $\alpha'(s)$.

Our goal is to understand what we can say about evolutes.

Involutes

Let $\alpha, \beta : (a,b) \to \mathbb{R}^3$ be regular curves. We say that $\beta$ is an involute of $\alpha$ when

1. $\beta(t)$ lies on the tangent line to $\alpha$ at $\alpha(t)$, and 
2. the tangents to $\alpha$ and $\beta$ at $\alpha(t)$ and $\beta(t)$ are orthogonal.

In this post, we will explore the basics of what one can say about an involute of a given curve.