Holonomy orbits of the snake charmer algorithm

A snake (of length L) is a (continuous) piecewise C1-curve S : [0, L] → R, parameterized by arc-length and whose “tail” is at the origin (S(0) = 0). Charming a snake consists in having it move in such a way that its “snout” S(L) follows a chosen C1-curve γ(t). The snake charmer algorithm, initiated in [Ha2] for polygonal snakes and developed in [Ro] in the general case, works as follows. The input is a pair (S, γ), where: (i) S : [0, L] → R is a snake of length L, (ii) γ : [0, 1] → R is C1-curve with γ(0) = S(L).