Two definitions of Evolute of a curve?

Question

Recall, when the tangents to a curve $\gamma$ are normal to another curve, the second curve is called an involute of $\gamma.$ In literature, there are two seemingly different dual notions for involutes.

Definition $1$
The evolute of a given curve $\gamma$ is another curve to which all the normals of $\gamma$ are tangent.

Definition $2$
Given a $\gamma$, another curve is called an evolute of $\gamma$ if it is an involute of the second.

With the second definition, and arc-length parametrization, an internet source shows that its evolute as $$\gamma(s)+\rho(s)N(s)+\rho(s)\cot\left(\displaystyle\int\tau ds+c\right)B(s).$$ But, for a plane curve, the first definition yields the "locus of all its centers of curvature" $$\gamma(s)+\rho(s)N(s)$$ as the evolute. This doesn't seems agree with the other for $\tau=0.$

• Are these two definitions actually inequivalent?
• If so, what is the correct terminology?

Also, I would like to see a reference discussing these types of constructions in the theory of curves.

Answer 1

EDITED

These definitions could be written far more clearly. So your first definition (definition 0?) should say that $\beta$ is an involute of $\gamma$ if for each $s$, $$\beta(s) = \gamma(s) + \lambda(s)T_\gamma(s) \qquad\text{and}\qquad T_\beta(s)\cdot T_\gamma(s)=0.$$ The first condition says $\beta$ lies on the tangent line of $\gamma$, and the second says their tangent vectors are orthogonal. (Note that I will use $s$ as an arclength parameter on $\gamma$, but it is far from one for $\beta$.) This is now the definition as I gave it in the comments.

Reversing letters so as to be less confusing, definition 1/2 says that $\gamma$ is an evolute of $\beta$ (i.e., $\beta$ is an involute of $\gamma$) if the principal normal lines of $\beta$ are tangent to $\gamma$, i.e., if $$\gamma(s) = \beta(s) + \mu(s)N_\beta(s) \qquad\text{and}\qquad N_\beta(s) = \pm T_\gamma(s).$$

A standard exercise is this: From definition 0, we deduce that $\beta$ is an involute of $\gamma$ if and only if $\beta(s)=\gamma(s)+(c-s)T_\gamma(s)$ for some constant $c$. Indeed, it is immediate from the Frenet equations that $T_\beta$ is $\pm N_\gamma$. Interchanging, by definition 2, $\beta$ is an evolute of $\gamma$ if and only if $T_\gamma = \pm N_\beta$. This gives you definition 1.

The derivation in the linked post is correct. There's no problem when $\tau=0$; you just take $c=\pi/2$. The careful statement — details matter! — is that there is some value of the constant $c$ for which the curve will be an evolute, not all.

Answer 2

Let me write what I gather so far, after spending almost a day on this problem.

Involute:
Given a regular parametric curve $\gamma$, an involute for $\gamma$ is another parametric curve $\gamma^*$ defined on the same interval such that
i) $\gamma^*(t)$ lie on $T(t)$ (tangent to $\gamma$ at $t$).
ii) $T(t)$ and $T^*(t)$ (corresponding tangent vectors) are orthogonal.

Immediately from this definition we have $\gamma^*(s^*)=\gamma(s)+\lambda(s)T(s)$ and $T^*\perp T,$ where $s, s^*$ are respective arc lengths. We can easily deduce that $T^*\parallel N$ and $$\gamma^*(s^*)=\gamma(s)+(c-s)T(s)$$ for any constant $c.$

Suppose we switch the roles of tangency and orthogonality, then we get my Definition $1$.
Given a $\gamma$, an evolute for it is another curve $\gamma^*$ defined on the same interval satisfying
i) $\gamma^*(t)$ lie on $N(t)$ (principal normal to $\gamma$ at $t$).
ii) $N(t)$ and $T^*(t)$ are parallel.

From here what we get is $\gamma^*(s^*)=\gamma(s)+\lambda(s)N(s)$ and $T^*\parallel N.$ Consequently, $\gamma$ must be a plane curve and we get the famous "locus of centers of curvature" equation: $$\gamma^*(s^*)=\gamma(s)+\rho(s)N(s).$$ However this definition is restricted as it works only for planer curves.

Now, we suppose the Definition $2$: $\gamma$ is an evolute for $\gamma^*$, if $\gamma$ is an involute for $\gamma^*$.
Then $\gamma(s)=\gamma^*(s^*)+\lambda(s^*)T^*(s^*)$ and $T\perp T^*.$
According to the second condition, $T^*$ must be a linear combination of $N, B$ (in the S-F frame of $\gamma$ at $s$). This leads to the representation $\gamma^*(s^*)=\gamma(s)+\alpha(s)N(s)+\beta(s)B(s)$ and ultimately the equation $$\gamma^*(s^*)=\gamma(s)+\rho(s)N(s)+\rho(s)\cot\left(\displaystyle\int_{s_0}^s\tau ds+c\right)B(s),$$ where $c$ is a constant.

It seems like the second definition generalizes the first one and hence is superior to that.