The complexity of tangent words

In a previous paper, we described the set of words that appear in the coding of smooth (resp. analytic) curves at arbitrary small scale. The aim of this paper is to compute the complexity of those languages.

💡 Research Summary

The paper investigates the combinatorial and geometric structure of “tangent words”, which are finite binary words that appear as cutting sequences of smooth (or analytic) planar curves when the underlying square grid is taken at arbitrarily small mesh size. By re‑encoding the four possible moves (right, up, left, down) as 0 and 1, the authors focus on two languages: T∞, the set of all tangent words generated by smooth curves, and Tω, the subset generated by analytic curves. Both languages are factorial (closed under taking factors) and extendable (any word can be prolonged).

A central tool is the desubstitution map δ, which removes one symbol from each maximal run of the non‑isolated letter (removing a 0 from each run of 0’s when the word contains no “11”, or a 1 from each run of 1’s when the word contains no “00”). Repeating δ until it cannot be applied yields a derived word d(w). The authors prove (Proposition 1) that w is a tangent word iff d(w) is accepted by a three‑state “diagonal” automaton, and w is a tangent‑analytic word iff d(w) is accepted by an eight‑state “non‑oscillating diagonal” automaton. This gives a purely combinatorial characterization of the two classes.

Geometrically (Proposition 2), a word w is tangent (resp. tangent‑analytic) precisely when, for any ε>0, it can be realized as the cutting sequence of a C¹‑curve that is ε‑close to a straight segment (resp. a C¹‑curve with nowhere‑zero curvature ε‑close to a straight segment). Thus the languages capture the local directional patterns of curves that are arbitrarily close to straight lines.

To compute the word‑complexity pₙ(L)=|{w∈L:|w|=n}|, the authors employ Julien Cassaigne’s theory of bispecial factors. A bispecial factor w is one for which all four extensions awb (a,b∈{0,1}) belong to L. Depending on how many of these extensions are present, a bispecial factor can be weak (2 extensions), ordinary diagonal (3 extensions), or strong (4 extensions). For both T∞ and Tω, there are no weak bispecial factors; every bispecial factor is strong. Moreover, each strong bispecial factor corresponds to a lattice segment from (0,0) to (p,q) with integer coordinates. If gcd(p,q)=1, the segment contains no interior lattice points, and the associated word of length p+q−2 is a strong bispecial factor for both languages. There are φ(n+2) such words of length n, where φ is Euler’s totient function.

When gcd(p,q)>1, let k be the number of interior lattice points on the segment. For tangent‑analytic words each interior point yields two possible “bends” (above or below), giving exactly 2·(n+2−φ(n+2)) strong bispecial factors of length n. For tangent words the situation is richer: each interior point can be bypassed in two ways, leading to 2^{k} possibilities. Summing over all divisors d of n+2 gives the total number of strong bispecial factors in T∞: \