What is the initial reason to define the evolute of a curve?

Question

The evolute of a curve is defined as the envelope of the normals or as the locus of the center of the osculating circle.

What is exactly "the envelope of the normals" ?

What is the reason to define such a concept ?

Answer 1

The evolute $\beta$ of a curve $\alpha$ is the locus of all of its centre of curvatures. So basically, an evolute is the envelope of the normals to the curve. $\beta$ is an evolute of $\alpha$ if $\alpha$ is an involute of $\beta$. Now, it can be proved that if $\beta$ is a unit-speed curve and if $\alpha$ is an involute of $\beta$ then \begin{equation*} \alpha = \beta + (c-s) \alpha' \end{equation*} where $c$ is a constant and $s$ is the arc-length. So $|\beta - \alpha|$ is a measure of arc-length of $\beta$ and hence $\alpha$ can be formed by unwinding a string from the curve $\beta$. Historically, the idea was given by Christian Huygens. It is said that he was trying to make a precise clock as the oscillation of a simple pendulam does not occur at equal intervals.He knew that the cycloid is the the curve for which the time taken by a particle sliding down under gravity is independent of its starting point. So if we take an inverted cycloidal arch bowl then the object reached the lowest point in exactly the same time, no matter from what height in the inner surface of the bowl, the object is released. So, now the problem reduces on finding that how does one get a pendulam to oscillate in cycloidal rather that circular arc. For this purpose he introduced the notion of involute of a plane curve. He showed that involute of a cycloid is a cycloid itself. So in prder to make the pendulam bob swing along a cycloid, the string neede to unwrap from a curve that is the evolute of a cycloid, so that the cycloid is the involute of the curve.\\ As to your first question recall that the envelope of a family of curves is a curve which is tangent to each member of the family at some point. Also an involute of a curve is lie on the tangent line of that curve and hence normals to a curve are all tangent to the evolute. That is why it is called the envelope of the normals.

Answer 2

The envelope of a family of lines is a curve which is tangent to all of the lines.

Physically, if the lines were light rays then the envelope would be a curve of extreme brightness. This is the envelope of light rays, or the caustic as it's called. I've attached a picture of a caustic.

The evolute is the envelope of the normals to a curve. That means that the evolute is tangent to all of the normal lines. Conversely, the tangent lines to the evolute are normal to the original curve at a corresponding point.

Note that the caustic in the image is not the evolute of the circle. The evolute of a circle is just its centre. An evolute is an example of a caustic, while not all caustics are evolutes. (Caustics include all envelopes to all families of lines. Evolutes are envelopes to families of normal lines.)

Generically, the evolute of a curve will be smooth with some isolates cusp points. The cusp points correspond to so-called vertices of the original curve. This is when the curve has higher order tangency with the osculating circle. In terms of the curvature, an ordinary vertex has $\kappa \neq 0$, $\kappa'=0$ and $\kappa'' \neq 0$.

If the evolute has a more degenerate singularity than an ordinary cusp, an arbitrarily small deformation of the curve will cause the singularity to break-up into several ordinary cusps.