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
collection.

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 *r*^{2}*f* / (*g*^{2} *-* *f*^{2}).)
Thus, **the Peaucellier linkage draws (at Q) the arc of a circle
without using the center of that circle.** If the lengths

**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.

*Mathematical
Aspects of the Peaucellier-Lipkin Linkage*