Let $\gamma$ be a smooth regular plane curve. We know that the curvature of $\gamma$ at $t$ is given by $||\gamma''(t)|| = \sqrt{x''(t)^2 + y''(t)^2}$, and the radius of the curvature at $t$ is $R(t) = ||\gamma''(t)||^{-1}$, provided that $||\gamma''(t)|| \neq 0$. Coming from CS background I'm always trying to think some/any application of the mathematical definitions/theorems I'm encountering, so naturally I'm trying to find some "use" for curvature. While the formulas allow us readily to compare which of two curves have a higher/lower curvature at $t_0$, I can't really find any other question in which the actual value of the curvature would be meaningful.
To further elaborate what I'm getting at: with derivative, we know the direction and the magnitude of the change of some multivariable function at a given point. Great!. Now we know how much and to what direction does our function, to which we assign meaning, change at some point. So what similar analogies do we have with curvature? I suppose we know how much the direction of our traverse changes at some point. Okay, what else/in which cases is this meaningful? You tell me. I suppose computer graphics/VR/AI research can easily use curvature as a tool for solving problems, but so far I haven't encountered anything that motivational (besides the mathematics itself).