Introduction to Linkages


A linkage, in computational geometry parlance, is a set of rigid edges connected by joints (i.e., vertices about which the edges can rotate.) There are various problems that surround linkages, such as:

How can the linkage avoid collision with other objects?
How can the linkage avoid collision with itself?
Can a given linkage (in the form of a closed polygon) be convexfied (moved in such a way so that all angles are made convex)?
Can a given linkage (in the form of an open polygonal chain) be made straight?
What kinds of curves can be drawn using one vertex of a linkage as a marker?
In how many different ways can a linkage fold?

Some problems with linkages are purely theoretical. Others can be applied in various ways, ranging from the design of mechanical objects (such as robot arms) to the design of drug molecules.

Here are some examples of simple linkages drawn with Cinderella. You can move the linkages by dragging the lighter-red points. Notice that the bars disappear when you try to stretch them farther than they are able to go.

Closed polygonal linkage and Open-chain linkage

The conformational space of a linkage is the "totality of all admissible positions in the Euclidean plane"(*). This Cinderella construction shows the conformational space of a very simple open-chain linkage. The conformational space in this example is composed of two circles whose radii correspond to the outstretched length of the linkage.


Linkage Links!

Animations of linkage convexifying, from Erik Demaine's page.
Information on a simple linkage: the four-bar linkage. Contains applets and instructions on building your own four-bar linkage, as well as theoretical aspects of four-bar linkage motion.
An applet demonstrating the use of linkages to draw curves.
More interesting linkage applets, from the Geometer's Sketchpad.
Parallel vs. crossed linkages: an example of their use in robot design.
Papers about linkages (among other subjects) available in PostScript format.
Description of a simplified model of protein folding, using open-chain linkages.


Now that you understand linkages, let's look at their applications to biology and chemistry:
Protein Folding

* "Configuraton Spaces of Mechanical Linkages," D. Jordan & M. Steiner, Discrete Computational Geometry 22 #2, 1999.