A Generalized Carpenters Rule Theorem for Self-Touching Linkages

📝 Abstract

The Carpenter’s Rule Theorem states that any chain linkage in the plane can be folded continuously between any two configurations while preserving the bar lengths and without the bars crossing. However, this theorem applies only to strictly simple configurations, where bars intersect only at their common endpoints. We generalize the theorem to self-touching configurations, where bars can touch but not properly cross. At the heart of our proof is a new definition of self-touching configurations of planar linkages, based on an annotated configuration space and limits of nontouching configurations. We show that this definition is equivalent to the previously proposed definition of self-touching configurations, which is based on a combinatorial description of overlapping features. Using our new definition, we prove the generalized Carpenter’s Rule Theorem using a topological argument. We believe that our topological methodology provides a powerful tool for manipulating many kinds of self-touching objects, such as 3D hinged assemblies of polygons and rigid origami. In particular, we show how to apply our methodology to extend to self-touching configurations universal reconfigurability results for open chains with slender polygonal adornments, and single-vertex rigid origami with convex cones.

💡 Analysis

The Carpenter’s Rule Theorem states that any chain linkage in the plane can be folded continuously between any two configurations while preserving the bar lengths and without the bars crossing. However, this theorem applies only to strictly simple configurations, where bars intersect only at their common endpoints. We generalize the theorem to self-touching configurations, where bars can touch but not properly cross. At the heart of our proof is a new definition of self-touching configurations of planar linkages, based on an annotated configuration space and limits of nontouching configurations. We show that this definition is equivalent to the previously proposed definition of self-touching configurations, which is based on a combinatorial description of overlapping features. Using our new definition, we prove the generalized Carpenter’s Rule Theorem using a topological argument. We believe that our topological methodology provides a powerful tool for manipulating many kinds of self-touching objects, such as 3D hinged assemblies of polygons and rigid origami. In particular, we show how to apply our methodology to extend to self-touching configurations universal reconfigurability results for open chains with slender polygonal adornments, and single-vertex rigid origami with convex cones.

📄 Content

arXiv:0901.1322v1 [cs.CG] 9 Jan 2009 A Generalized Carpenter’s Rule Theorem for Self-Touching Linkages Timothy G. Abbott∗† Erik D. Demaine∗‡ Blaise Gassend§ Abstract The Carpenter’s Rule Theorem states that any chain linkage in the plane can be folded con- tinuously between any two configurations while preserving the bar lengths and without the bars crossing. However, this theorem applies only to strictly simple configurations, where bars inter- sect only at their common endpoints. We generalize the theorem to self-touching configurations, where bars can touch but not properly cross. At the heart of our proof is a new definition of self-touching configurations of planar linkages, based on an annotated configuration space and limits of nontouching configurations. We show that this definition is equivalent to the previously proposed definition of self-touching configurations, which is based on a combinatorial descrip- tion of overlapping features. Using our new definition, we prove the generalized Carpenter’s Rule Theorem using a topological argument. We believe that our topological methodology pro- vides a powerful tool for manipulating many kinds of self-touching objects, such as 3D hinged assemblies of polygons and rigid origami. In particular, we show how to apply our methodology to extend to self-touching configurations universal reconfigurability results for open chains with slender polygonal adornments, and single-vertex rigid origami with convex cones. ∗MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA, {tabbott,edemaine}@mit.edu †Partially supported by an NSF Graduate Research Fellowship and an MIT-Akamai Presidential Fellowship. ‡Partially supported by NSF CAREER award CCF-0347776, DOE grant DE-FG02-04ER25647, and AFOSR grant FA9550-07-1-0538. §Exponent Failure Analysis Associates, 149 Commonwealth Drive, Menlo Park, CA 94025, USA, blaise.gassend@m4x.org. Work done while at MIT. 1 1 Introduction In the mathematics of geometric folding [O’R98, Dem00, DD01, DO05, DO07], a common ideal- ization is to model the underlying real-world object—a mechanical linkage, robotic arm, protein, piece of paper, or another object or surface—as having zero thickness. The rods or bars that make up a linkage become perfect mathematical line segments of fixed length; the joints or hinges that connect them become mathematical points; a piece of paper can be folded repeatedly ad infinitum. While these idealizations are not entirely realistic (see [Gal02]), the zero-thickness model has led to a wealth of powerful theorems that rarely abuse the lack of thickness and are therefore practical. nontouching noncrossing self−crossing self−touching Figure 1: The different types of configurations. Almost all forms of folding forbid folding ob- jects from crossing, matching a natural physical constraint, but at the same time allow folding objects to touch. Figure 1 illustrates the dis- tinction between touching and crossing. For ex- ample, overlapping multiple layers of paper en- ables origamists to form arbitrarily complicated shapes, both in practice and in theory [DDM00]. Touching is easy to model for objects with positive thickness: allow the boundaries, but not the interiors, to intersect. But in the zero-thickness model, formally distinguishing between touching and crossing is difficult. In particular, when two portions of the object overlap, the geometry alone is insufficient to distinguish which portion is on top of which. The approach taken so far to resolving the ambiguity is to express the information missed by the geometry with additional combinatorial information. A simple example is map folding of an m × n grid of squares [ABD+04]. In this context, the geometry of the squares is completely determined, independent of the folding: in any successful folding that uses all the creases, all of the squares will end up on top of each other, with orientations specified by a checkerboard pattern in the grid. The folding itself can be specified by a purely combinatorial object: the permutation of the panels that describes their total order in the folding. The challenge is to determine what constraints on this combinatorial object correspond to the paper not self-crossing. A generalization of this approach is essentially the one taken by [DDMO04, DO07] for defining general origami. There are two concerns with this type of approach. First, how do we know that the combinatorial definition corresponds to the intended meaning of self-touching configurations? The combinatorial definitions inherently lack geometric intuition, so it is hard to “feel” that they are correct, even though we believe they are. Second, how do we manipulate these definitions to prove interesting theorems? The complexity of the definitions makes them hard to use. While some problems were successfully attacked in [CDR02, DDMO04], many other problems about self-touching configurations remain open. An alternate, equivalent definition would give

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