>>13929610>>13929610We can do it in full generality if we're in parametric equations
Let be any parametric curve. You want to make your curve squiggly with a sine whose direction of oscillation is perpendicular to the direction of the curve
The direction of the curve is parallel to the vector . The normalized direction vector is . The normalized direction vector 90 degrees counterclockwise from this is . Therefore if you want your smaller oscillation to have amplitude and wavelength , your new parametric equation is
For the case where your original parametric curve is a standard sine wave, this becomes
https://www.desmos.com/calculator/qsffvecoqkHowever this solution has the drawback that the little wavelengths are evenly spaced with respect to the x-axis and not with respect to actual arc of the big curve, making the picture look rather jagged, especially as k gets closer to 1. Patching this up so that the wavelengths are even on the curve will probably require some kind of arc length argument which may or may not have a closed-form solution. Hell, maybe it cleans up into something nice, although I doubt it
