HOLONOMY ORBITS OF THE SNAKE CHARMER ALGORITHM JEAN-CLAUDE HAUSMANN and EUGENIO RODRIGUEZ

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 in...

The space of clouds in an Euclidean space Jean - Claude HAUSMANN and Eugenio RODRIGUEZ

We study the space Nm d of clouds in Rd (ordered sets of m points modulo the action of the group of affine isometries). We show that Nm d is a smooth space, stratified over a certain hyperplane arrangement in Rm. We give an algorithm to list all the chambers and other strata (this is independent of d). With the help of a computer, we obtain the list of all the chambers for m ≤ 9 and all the str...

The space of clouds in an Euclidean space

We study the space Nm d of clouds in Rd (ordered sets of m points modulo the action of the group of affine isometries). We show that Nm d is a smooth space, stratified over a certain hyperplane arrangement in Rm. We give an algorithm to list all the chambers and other strata (this is independent of d). With the help of a computer, we obtain the list of all the chambers for m ≤ 9 and all the str...

Geometric descriptions of polygon and chain spaces

We give a few simple methods to geometically describe some polygon and chain spaces in R. They are strong enough to give tables of m-gons and m-chains when m ≤ 6.

Mod Two Homology and Cohomology