S35: Brief Description of the Mathematics Behind the Peaucellier-Lipkin Linkage
By Daina Taimina and David W. Henderson
This linkage starts with the question: What is straight?
When using a compass to draw a circle, we are not starting with a figure we accept as circular; instead we are using a fundamental property of circles that the points on a circle are at a fixed distance from the center. In other words we are using the definition of a circle.
Is there a tool (serving the role of a compass) that will produce a motion along a straight line?
o If, in this case, we want to use Euclid's definition (A straight line is a line which lies evenly with the points on itself) -- it won't be of much help.
o One can say: We can use a straightedge for constructing a straight line! Well, how do you know that your straightedge is straight? How can you check that something is straight?
This question was important for James Watt when he was thinking about improving steam engines he needed a mechanism to convert circular motion into straight-line motion and vice versa. In 1784 Watt found a solution (which he called "parallel motion") that was developed in several different configurations. His "parallel motion" was a practical (and aesthetic) solution to his problem, but it produced only approximate straight-line motion it actually produces a stretched out figure eight. The Reuleaux models S24, S25, S37, S38 are examples of various configurations of Watt’s approximate straight-line motion linkages. We suggest that the reader investigate these Watt’s linkages by exploring their movies and simulations.
Figure 1. Model of Watt's steam engine from the KMODDL
The “parallel motion” linkage is in the top left corner.
Mathematicians were not satisfied with the approximate solution and worked for almost a hundred years to find linkages that would produce exact straight-line motion. See the Reuleaux models, S05, S06, S07, S30, S32, for other examples of these approximate solutions. (See How to Draw a Straight Line for further discussions of the relevant history.)
A linkage that draws an exact straight line was not found until 1864-1871 when a French army officer, Charles Nicolas Peaucellier (1832-1913), and a Russian graduate student, Lipmann I. Lipkin (1851-1875), independently developed a linkage that draws an exact straight line.
Figure. Peaucellier-Lipkin linkage
Why does the Peaucellier-Lipkin linkage draw a straight line? We suggest the reader first view the mechanism in motion. One can derive, as an exercise in analytic geometry, that the point Q, on the far right, will always lie along a straight line but this does not answer the mathematical question: Why does it draw a straight line? Especially difficult is to see any relationship with the usual meanings of ‘straightness’ ‘straight’ as ‘shortest’ or ‘straight’ as ‘symmetric’ Is there perhaps a different third meaning of straightness that is operative here?
In the “inversor” (the links labeled d and s) the points P and Q are inverse pairs with respect to a circle with center C and radius r= -- analytically, this means that
Here, the crucial property of circle inversion is that it takes circles to circles. See inversion-circles Java Simulation for simulation of the inversion of circles in a circle.
(For details of circle inversions, see Mathematical Aspects of the Peaucellier-Lipkin Linkage or [Henderson/Taimina, Experiencing Geometry, 2005].)
After experiencing the motion of the linkage we see that P is constrained (by its link to the stationary B) to travel in a circle around B and thus Q must be traveling along the arc of a circle the radius and center of this circle is varied by changing the position of the fixed point B the length of the link BP. (In Mathematical Aspects of the Peaucellier-Lipkin Linkage it is shown that the radius of the circle is r2f / (g2 - f2).) Thus, the Peaucellier linkage draws (at Q) the arc of a circle without using the center of that circle. If the lengths g (CB) and f (BP) are equal then the circle on which P moves goes through the center C; and, since points near C are inverted to points near infinity, the circle that Q lies on must go through infinity!
How can a circle go through infinity? Answer: Only if the circle has infinite radius. A circle with infinite radius (and thus zero curvature) is a straight line. We now have a third meaning for straight line; and the Peaucellier-Lipkin linkage is a tool for drawing a straight line using this meaning.
References for more details:
David W. Henderson and Daina Taimina, Experiencing Geometry: Euclidean, Non-Euclidean With Strands of History, Upper Saddle River, NJ: Pearson Prentice-Hall, 2005.