Constraining Plane Configurations in Computer-Aided Design: Combinatorics of Directions and Lengths

Abstract

Configurations of points in the plane constrained by directions only or by lengths alone lead to equivalent theories known as parallel drawings and infinitesimal rigidity of plane frameworks. We combine these two theories by introducing a new matroid on the edge set of the complete graph with doubled edges to describe the combinatorial properties of direction-length designs.

Rigidity and scene analysis

“…If angles and incidences are added, even the problems of "generic rigidity" of constraints are unsolved (and perhaps not solvable in polynomial time). However, special designs, mixing lengths, distances of points to lines, and trees of angles have been solved, using direct extensions of the techniques and results for plane frameworks and plane parallel drawings [SW99].…”

Operations Preserving Global Rigidity of Generic Direction-Length Frameworks

“…In Section 2 we will show, for completeness, that infinitesimal rigidity (defined in the next section) is a sufficient condition for rigidity in mixed frameworks, and that the two conditions are equivalent for generic mixed frameworks. The above mentioned characterization of rigid mixed graphs is given in terms of infinitesimal rigidity in [14] and we need this equivalence to justify our statement of their result, Theorem 1.2. In Section 3 we introduce quasi-generic frameworks and derive a result about the field extension Q(p) of a quasigeneric rigid framework (G, p) which will be key to our main results on extensions.…”

Continuously Flattening Polyhedra Using Straight Skeletons

“…(In this case, their shared crimp will have width |wi − wj|/2 after one second of motion.) This orthogonality condition means precisely that the vectors wi define an orthogonal embedding of the dual graph of G, and in particular that G is a spiderweb [29]. So the spiderweb constraint does indeed arise naturally.…”