If we have a function $f(x)$, what is the function $g(x)$ where every line between $f(x)$ and $g(x)$ and perpendicular to $f(x)$ is equal? (If there is a better way to phrase this, let me know in the comments.)
So my idea was that the slope of the line perpendicular to $f(x)$ equal $\frac{b}{a}=-\frac{1}{\frac{d}{dx}f(x)}$. Let the distance between $f(x)$ and $g(x)$ along that line be $r$. And $a^2+b^2=r^2$. So when we solve individually for $a$ and $b$ in the first equation, substitute in the second, then solve for $a$ and $b$ again, we get $$a=\frac{r}{\sqrt{1+\left(\cfrac{1}{\frac{d}{dx}f\left(x\right)}\right)^2}},\\b=\frac{r}{\sqrt{1+\left(\frac{d}{dx}f\left(x\right)\right)^2}}$$ So for a value $t$ we would have $$x= t+\frac{r}{\sqrt{1+\left(\cfrac{1}{\frac{d}{dt}f\left(t\right)}\right)^2}}$$ and $$g(x)=f(t)-\frac{r}{\sqrt{1+\left(\frac{d}{dt}f\left(t\right)\right)^2}}$$ So I guess if we solved for $t$ in the top equation and plugged that into the bottom equation we would have our function. Solving for $x$ has proved to be difficult for simple functions like $f(x)=x^2$. I'm wondering if there is a way to solve it for a general power of $x$ or if there are other interesting functions we could use. Maybe there are better routes to go and I would love to see what those are.
"A parallel of a curve ... can also be defined as a curve whose points are at a fixed normal distance from a given curve," says Wikipedia. Many examples are given there, and a formula in case the original curve is given in parametric form, otherwise references to algorithms and approximations.