The Fabricius-Bjerre formula is a rather isolated and peculiar result which gives a combinatorial forumula relating key geometric properties of a curve. If we assume the curve to be regular enough not to run into issues, we can consider its finite number of inflexion points, self-crossings and distinguish two types of bitangents to the curves. Namely, when a tangent to our curve is tangent to the curve at two points, either the curve lies on one side of the tangent, or it lies on either side of the bitangent. Calling the former type positive bitangents and the latter negative, Fabricius-Bjerre proved the following elegant formula:

$|\{Positive Bitangents\}| = |\{Negative bitangents\}| + |\{Self-Crossings\}| + \frac{1}{2} |\{Inflection points\}|$

Ignorant of this formula, I became curious about tangent polygons inscribed in a plane curve (bitangents are thus the particular n=2 case) and noticed the following fact:

(Observation). As soon as a plane curve has two inflections points digging a non-convex “well” into the curve, there will be an infinity of tangent polygons inscribed in the curve, and an $n$ -gon for each $n$ . Moreover, the vertices of these polygons form a Cantor set.

(See this write-up I typed in my first year of undergrad for more details)

I then started focusing on tangent triangles and hoped for a combinatorial formula that would generalise Fabricius-Bjerre’s formula to tangent polygons inscribed in a curve. There is indeed also a sensible way to define positive and negative polygons and it almost seemed like the following conjecture was true:

(False Conjecture, but Somehow Almost True). $|\{Positive Polygons\}|=|\{Negative Polygons\}|$

(See this short/rough write-up for more details)

After discretizing the problem and reducing the problem to polygonal curves, I coded up a program in matlab to generate random curves and count tangent triangles. Remarkably, although I can show how to make the gap between the two quantities arbitrary, it really does seem like for most generic, normal looking curves, the two quantities are almost always equal or differ by at most 1 or 2 no matter how complicated the curves. And when thinking about it a bit deeper, a similar argument to the original proof of Fabricius-Bjerre can somewhat explain this. But the analysis of what the correction term might be is quite intricate. I’m still hoping to dedicate more time to this problem and figure it out though! Oh and by the way, there are fascinating connections between Fabricius-Bjerre’s formula and Arnold’s invariants for plane curves and Uli Wagner also figured out a connection to the $k$ -set problem using the formula and the crossing lemma to bound the self-crossings.

Counting Tangent Polygons inscribed in Plane Curves