Though not obvious from the title, this paper is concerned with 3D geometric constraints, as arise in, say, molecular modeling. This is definitely not a straightforward generalization of the commonly treated 2D case. The authors give many reasons for this: perhaps the clearest is that “a combinatorial characterization of rigidity is known for 2D realizations of graphs but not for 3D ones”; another important one is that a set of n constraints may have exponentially many (in n) solutions. The authors therefore do not propose to find all solutions by default.

The authors propose instead to start with a (manual) sketch of the solution and transform it into an actual solution satisfying all of the constraints (lengths, coplanarity, and so on). This approach is clearly crying out for homotopy-based methods. The authors claim (and the experimental data presented certainly support this) that taking account of this setting, and the sort of degeneracies that can occur, is far more efficient than general homotopy solutions (they used Hom4ps [1] as a comparison).

The authors note that their method can cope efficiently with up to about 40 constraints, and that “in order to make it usable in an industrial product, it has to be combined with a decomposition method to tackle bigger problems.” This is an interesting challenge.

The editorial process for this paper was less than perfect: several times (for example, “one by distance constraint”), I was forced to translate word-for-word back into French in order to understand the meaning.