Can we estimate how the curvature of a function changes if "bent" over another function?


  • $y_1(s)$ - is sinusoidal function, e.g., $a\cdot \sin(b\cdot s), \; s \in [0, L]$
  • $y_2(t)$ - is a circular curve, e.g., $\bigl(r-r\cdot \cos(\frac{L\cdot t}{r}), r\cdot \sin(\frac{L\cdot t}{r})\bigr), \; t \in [0, 1]$

example plot

Can we safely assume that the curvature of the orange line will be $\sim 1/r$ higher than the one of the original sinusoid?

For a given example a numerical estimation (using the convolution method) gives an increase of approximately $1/r$. I can't find a way to derive the curvature analytically, though, so I don't know if it is a luck or can be generalized.

  • 1
    $\begingroup$ You need to start by parametrizing this curve. I expect you're doing something like $$f(t) = y_2(t) \pm ry_1(t)N(t),$$ where $N$ is the principal normal of the base curve. Now you can calculate (using the formula for curvature of a non-arclength-parametrized curve). Note that your $y_2$ is a parametric curve and your $y_1$ is a scalar function. Note also that your $t$ parameters to not align, but you can fix that by scaling $t$ in the $y_1$ slot. $\endgroup$ Apr 2 at 17:25
  • $\begingroup$ This is a generalization of offset curve or parallel curve. Have you looked into this special case (where $y_1$ is a constant)? $\endgroup$ Apr 2 at 17:26
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    $\begingroup$ By the way, your title is totally misleading. In no way is this a product of two functions. $\endgroup$ Apr 2 at 17:27
  • $\begingroup$ @TedShifrin, thanks for replying. I've modified the equation slightly so it corresponds to the image attached. Deriving the x(t) and y(t) is not complicated (thus the curve above), but to calculate curvature I need to express it in terms of the $s$ - elementary length along the curve, and this requires some integration which in general not always be possible. About the title: how would you call it? $\endgroup$
    – Roman Zh.
    Apr 2 at 20:25
  • 3
    $\begingroup$ You do not need an arclength parametrization. You can use the chain rule in lieu of that. See pp. 12-14 (in particular, Proposition 2.2) of my differential geometry text, linked freely in my profile. There is no official name for such a curve, although it's roughly a 2-D version of a helix; I guess I would call it a plane curve oscillating about another. $\endgroup$ Apr 2 at 20:49


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