# Evolutes and Extensions

Wikipedia provides a nice definition of evolutes for parametric curves. One definition is that the evolute of some curve $$\gamma$$ is the envelope of the curve's normals. What we will consider is the envelope of the lines off by a fixed angle $$\theta$$ to the normals of $$\gamma$$.

## Definition/Derivation

For some $$U \subseteq \mathbb{R}$$, let $$f : U \rightarrow \mathbb{R}$$ be some $$C^2$$ function, and let $$\gamma(t) = (t, f(t))$$. At any point $$(x_0, f(x_0))$$, the normal to $$\gamma$$ is given by the line $$y - f(x_0) = \frac{-1}{f'(x_0)}(x - x_0)$$. For a fixed angle $$\theta$$, the line forming an angle of measure $$\theta$$ from the normal (in the counterclockwise direction) can be constructed as follows. The vector $$(1, f'(x_0))$$ is parallel to the normal of $$\gamma$$ at $$x_0$$, so we simply apply the appropriate rotation matrix to this vector, and construct the line with that slope going through the point $$(x_0, f(x_0))$$.

$$\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin\theta) & \cos(\theta) \end{bmatrix} \begin{pmatrix} 1 \\ f'(x_0) \end{pmatrix} = \begin{pmatrix} \cos(\theta) - f'(x_0)\sin(\theta) \\ \sin(\theta) + f'(x_0)\cos(\theta) \end{pmatrix}$$

Rescaling this vector so that the first entry is one yields the following line.

$$y - f(x_0) = \frac{\sin(\theta) + f'(x_0)\cos(\theta)}{\cos(\theta) - f'(x_0)\sin(\theta)}(x - x_0)$$

Then we can explicitly compute the intersection of the two lines

$$y - f(x_0) = \frac{\sin(\theta) + f'(x_0)\cos(\theta)}{\cos(\theta) - f'(x_0)\sin(\theta)}(x - x_0) \quad \text{and} \quad y - f(x_0 + h) = \frac{\sin(\theta) + f'(x_0 + h)\cos(\theta)}{\cos(\theta) - f'(x_0 + h)\sin(\theta)}(x - x_0),$$

as the following

$$f(x_0) + \frac{\sin(\theta) + f'(x_0)\cos(\theta)}{\cos(\theta) - f'(x_0)\sin(\theta)}(x - x_0) = f(x_0 + h) + \frac{\sin(\theta) + f'(x_0 + h)\cos(\theta)}{\cos(\theta) - f'(x_0 + h)\sin(\theta)}(x - x_0)$$ $$\Rightarrow f(x_0 + h) - f(x_0) = (x - x_0) \left[ \frac{\sin(\theta) + f'(x_0 + h)\cos(\theta)}{\cos(\theta) - f'(x_0 + h)\sin(\theta)} - \frac{\sin(\theta) + f'(x_0)\cos(\theta)}{\cos(\theta) - f'(x_0)\sin(\theta)} \right]$$

where

$$\frac{\sin(\theta) + f'(x_0 + h)\cos(\theta)}{\cos(\theta) - f'(x_0 + h)\sin(\theta)} - \frac{\sin(\theta) + f'(x_0)\cos(\theta)}{\cos(\theta) - f'(x_0)\sin(\theta)}$$ $$= \frac{(\sin(\theta) + f'(x_0 + h)\cos(\theta))(\cos(\theta) - f'(x_0)\sin(\theta)) - (\sin(\theta) + f'(x_0)\cos(\theta))(\cos(\theta) - f'(x_0 + h)\sin(\theta))}{(\cos(\theta) - f'(x_0 + h)\sin(\theta))(\cos(\theta) - f'(x_0)\sin(\theta))}$$ which reduces nicely via some trigonometric identities to $$= \frac{f'(x_0 + h) - f'(x_0 + h)f'(x_0)\cos(\theta)\sin(\theta) - f'(x_0) }{(\cos(\theta) - f'(x_0 + h)\sin(\theta))(\cos(\theta) - f'(x_0)\sin(\theta))}$$

Note: Although we have only done the work for functions of $$x$$, morally speaking there is nothing stopping us from doing the same for parametric curves. Indeed, you could just partition some parametric curve into components, each of which being a real-valued function, and construct our extended evolute for each component. I've not looked into this for some years now, but last I did, I was unable to actually derive a non-piecewise closed form for parametric curves.