I've been experimenting with the curvature of parametric curves on Desmos, and have gotten the following results. I cant really explain my question with words, so here's a bit of an introduction to my work which will provide some context.
I defined the parametric curve in Desmos with $\tau_2(x)=(x_1(x),y_1(x))$ for any choice of (at least 3-times differentiable) functions $x_1(x)$ and $y_1(x)$. This graphs nothing, so one must define $\tau_1(x)=\tau_2(\phi x)$ where $\phi>0$ is any constant represented by a slider. This again graphs nothing. Enter $\tau_1(t)$ and specify that $-1≤t≤1$ to see the parametric curve. Slide $\phi$ and watch the curve change in length/domain.
Next define $d(x,y)=\sqrt{x^2+y^2}$, and $u(x,y,z)=\frac{x}{d(y,z)}$. Then define $t_1$ as any potential value for $t$. $$\tau_2(t_1)$$ gives the parametric curve/function thingy evaluated at $t=t_1$. Everything will be centered at this point.
Then $T_1(x)=u(x_1'(x),x_1'(x),y_1'(x))$ and $T_2(x)=u(y_1'(x),x_1'(x),y_1'(x))$, and $T(x,y)=(T_1(x)y+x_1(x),T_2(x)y+y_1(x))$. $$T(t_1,t), 0≤t≤1$$ gives the unit-length tangent segment to the point $\tau_2(t_1)$. You can do more with this but I won't get that into it as it is not very relevant. getting the visual of $T(t_1,t)$ is nice, but not super necessary to the construction of my question.
Next $N_1(x)=u(T_1'(x),T_1'(x),T_2'(x))$ and $N_2(x)=u(T_2'(x),T_1'(x),T_2'(x))$. Then $N(x,y)=(N_1(x)y+x_1(x),N_2(x)y+y_1(x))$. And $N(t_1,t)$ for $0≤t≤1$, gives the normal unit-length segment to $\tau_2(t_1)$.
Then the real meat of the question: curvature. The radius of curvature is $$r(x)=\frac{d(x_1'(x),y_1'(x))}{d(T_1'(x),T_2'(x))}$$
From this we define the radius of curvature segment-function $R(x,y)=(r(x)N_1(x)y+x_1(x),r(x)N_2(x)y+y_1(x))$. Thus $$R(t_1,t), 0≤t≤1$$ gives the actual radius segment. And $R(t_1,1)$ gives the center of the circle of curvature, or whatever it's called.
Finally: My actual questions
Consider the curve given in Desmos by $R(\phi t,1), -1≤t≤1$. What is this curve called? Is it important? What about $R(\phi t,i), -1≤t≤1$, where $i$ is the list $[0,0.1,...,1]$? Is there a non-Desmos equivalent of these curves?
Edit: Here's a graph of everything I described.
Just to turn Rahul's comment into an answer, it seems like you're thinking of the evolute of a curve, which is the locus of the curve's centers of curvature.