Category Archives: curve-fitting

I have experimented with integrals of exp(i·f(t)) where ”f(t)” is a polynomial. I express these integrals as Taylor series, which are not hard to generate. Now I’d like to try f(t) = a·t + Σbk·sin(k·t) and I have a horrid … Continue reading →

Many of the curves in this chart have some unsightly wiggles. That’s because, when a function of degree 2 or higher tries to approximate a piecewise constant, it tends to go back and forth across the target. So here instead … Continue reading →

I wrote: each curve hits alternate dots: first exactly, then with offsets pushing it toward the other curve. I don’t think I’ve mentioned here how the offsets work.

For context, see past posts in the curve-fitting category that I just created. To recap: The curves I’ve been drawing are the paths made by a point moving at constant speed at an angle which is a piecewise quadratic function … Continue reading →

Here, each curve hits alternate dots: first exactly (above), then with offsets pushing it toward the other curve. Below is the result of eight iterations. With enough iterations, the top of ‘s’ eventually gets a more symmetrical arch, as the … Continue reading →

(Previously: 2014, 2011, 2010; also, less closely related, 2015) I tried to smoothen a stroke by shifting each dot toward the Euler spiral (aka clothoid, aka Cornu spiral) determined by its four nearest neighbors. That didn’t work so well: small … Continue reading →

One thing I noticed in that last series of charts is that more than one degree of discontinuity doesn’t help: the best-looking curves are mostly on the diagonal, where only the last nonzero derivative is discontinuous. Here, therefore, are those … Continue reading →