**The election**
I'm still in a state of shock. I think things are going to get quite a
bit worse before they get better again. I also think this outcome is
likely to be bad for free software in many different ways, including
strong anti-American sentiment around the world, shoring up of
corporate power, and a general devaluation of the public interest as
an important factor in political decision-making.

**Curves: Ikarus**

Ikarus
is a near-legendary system for digitizing outline curves for fonts. I
had a copy of Karow's 1987 book explaining the system (complete with
FORTRAN source code, no less), but somehow managed to let it go. I've
been meaning to get back to it for a while.

A month or so ago, I wrote that I was pretty sure that Ikarus used
Hermite splines. In response, haruspex
disagreed and said it was circular arcs. I was dubious because
assigning one circular arc between each pair of adjacent knot points
yields an overly rigid numerical system.

It turns that we were both right. The actual Ikarus system uses a
hybrid approach. The first step is to fit a cubic spline
to the knot points (with weighting based on distance between knots, to
improve cases when the knots are not evenly spaced). However, from
this phase, only the tangents are preserved; the spline curves
themselves are discarded.

At this point, you have position (the original input) and tangent
constraints for each knot. Those constraints uniquely determine a biarc for
each segment between two knots. Since a biarc is made up of two
circular arcs (tangent continuous at the join), you have an extra
degree of flexibility that avoids the rigidity I was worried about.

So now I can state clearly the similarities and differences between
Ikarus and my Cornu-based approach. They both try to fit a smooth
curve through the input points, and both work in two passes, the first
of which finds the tangents and the second of which draws instances of
the primitive geometric curve. However, I use a Cornu spiral segment
where Karow uses a biarc, and I choose the tangents so that the
curvature of the *final* curve is continuous. Ikarus, by
contrast, chooses the tangents so that the curvature of the
intermediate cubic polynomial is continuous. By nature, you can't make
biarcs continuous in curvature, but Karow's solution certainly counts
as a rough approximation of that goal.

I need to make a visual illustration of the differences, but I expect
the Cornu approach to be better in all respects. Cornu spiral
segments, while resembling biarcs (and having precisely the same
control parameters) have smoother variation of curvature. Similarly, I
expect the tangent solution to be more accurate, not least because it
guarantees curvature continuity of the final curve. When there is a
large number of points (Ikarus recommends one knot for every 30
degrees of arc, and the illustrations in Karow's book seem to be even
finer), the two solutions should be nearly identical. However, I think
you can get away with about half as many points with the Cornu spline
without sacrificing smoothness. That should make it easier to maintain
high quality while editing and otherwise distorting, as well as the
obvious savings in time because you don't need to enter as many
points.

Btw, the Ikarus book also clears up my hazy memories about previous
spiral approaches to curves. I thought there was a historical spiral
format due to Coueignoux, but the one I was thinking of is actually by
Purdy and McIntosh. A survey
paper by Lynn Ruggles has all the details for the curious, or read
the patent.