"Never mistake motion for action." - Ernest Hemingway
Soon I hope to have a tutorial document on this subject
illustrated
with animation. However, in the mean time you can read about the topic
in an early survey paper I wrote or play with the interactive Java
applets
found on some of the links listed below.
Godfried T. Toussaint, "Movable
separability
of sets," in Computational Geometry, Ed., G. T. Toussaint, North-Holland,
1985, pp. 335-375. This paper is a tutorial survey of
research results
obtained before 1985 related to disassembly and interlocking puzzles in
two and three dimensions. This area is sometimes called local motion
planning.
The research described in the above papers is in fact
concerned
with the general theory of interlocking polygons in the plane and
polyhedra
in space. Therefore the analysis and design of interlocking puzzles
such
as burrs is closely related to local motion planning as well as
assembly
and disassembly in manufacturing. The literature on burr-puzzles is
very
helpful for thinking about and visualizing these kinds of geometric
problems.
Links to related sites:
The Sofa Problem: The
sofa problem asks for the largest (maximum-area) shape that can be
moved
around a corner in a corridor. This is still an open problem but upper
and lower bounds on the solution exist. See the nice
interactive applet where you can move several sofas of different
lengths.
Separability, and mobility in general, of
objects
are of
course intimately related to immobility which, in turn, is the central
issue of grasping
and fixturing problems in robotics and manufacturing. Check
out the postscript document below.
Godfried
T. Toussaint, "All convex polyhedra can be clamped with
parallel jaw grippers," Computational
Geometry: Theory and Applications,
vol. 6, 1996, pp. 291-302. (with P. Bose and D. Bremner)
IBM's
Burr Puzzle Page contains lots of information on the history and
design
of burr puzzles as well as interactive Java applets to take apart as
well
as design your own 3D puzzles.
Jack
Snoeyink and Jorge
Stolfi
have obtained some beautiful results on taking apart a set of objects
with
only two hands. Their theorem inspired a sculpture presently hanging in
the atrium of the Department of Computer Science at the University of
British
Columbia. Jack's page contains links to a description of the problem, a
paper containing the mathematical results and photographs of the
sculpture.
The problem of determining if objects in space
are
interlocked
or not is closely related to problems in the theory of knots
and links although the latter are of course more topological than
geometric.
Linkages, Rigid and
Movable
Polygons:
The Four-Bar Linkage:
Godfried T. Toussaint, "Simple
proofs
of a geometric property of four-bar linkages," American
Mathematical
Monthly, Vol. 110, No. 6, June-July 2003, pp. 482-494.
