relax

This curve-fitting thingy is one of several projects on which I’ve made progress in rare fits over several years. It ran into two big snags. I haven’t found how to determine which gridpoints are within the pen-width of a blending arc; two methods that ought to work don’t. (What would help: tutoring in drawing pictures in a MacOS display, so that I might have a better idea where they go wrong. And a pony.)

The other snag is this: For each pair of arcs, there is an infinite family of blending arcs; how to choose the osculation points to minimize rapid changes in curvature, while meeting the gridpoint constraints? ( . . more . . )

it’s all connected

My old calculus book gives a formula for the curvature of a parametric arc in the plane — that is, an arc defined by two functions (x(t),y(t)) of one variable. For thirty years I didn’t think about the derivation of that formula. Just now it hit me (and I did the algebra to confirm) that, in terms of the complex plane (z=x+iy), the curvature formula is equivalent to

Im(z″/z′) / |z′|

This should improve my cubic approximations to transcendental curves.

blending curves

I made some progress on an old project: to make outline fonts based on some favorite old bitmap fonts, by automatic fitting of smooth curves to the sequences of dots.

(The image above is in Scalable Vector Graphic format. If you see nothing, you may – dare I say it – need to update your browser. It works in Firefox 5, Safari 5, Chrome 12, Kindle.) ( . . more . . )

toward a graceful imitation of the crude

I’ve long had an idea to design “outline” typefaces which, at appropriate low resolution, would mimic certain bitmap fonts that have sentimental resonance.

The orange discs are the original dots, of course. The blue arcs are least-squares fits (linear, quadratic) to subranges of the dots. The arcs are blended with a weighting function that favors longer arcs, as well as the middle of each arc. Finally, the stroke is thickened by adding ±*i*/2 to the parametric variable.

This is the first version in which the stroke-ends are neither brutally stiff nor (in some cases) grotesquely exuberant. I don’t know yet whether the lumpiness, here and there, reflects ~~a flaw~~ an opportunity to improve the blending function or a limitation of the cubic splines used to simplify the final curve.

(I previously made a TrueType version of Apple’s “Los Angeles” font, by a much more *ad hoc* approach.)

Takana go goth

See Takana. The 306 figures shown there can be reduced to 45 by rotation and reflection. I fitted a polynomial curve to each partial path, and superimposed them.