by
D. Hembree

A topic that is never treated in high school geometry (or physics) is that of mechanical linkages or mechanisms. The topic is actually one of mechanical engineering, but provides a rich field for geometric exploration. A dynamic geometry software package such as Geometer's Sketchpad can be used to create and explore the possibilities and limitations of many mechanical devices.
It is easy to believe that the "electronic age" has surplanted the need for mechanical devices, but consider:

the sliding tray on your computer's CD drive opens by mechanical means, your printer, copy machine, fax, or scanner are full of gears and linkages, the steering and suspension on your car has changed very little since its conception, linkages operate sewing machines, shutters on film cameras, piano keyboards, construction machinery, and the parking lot arms that keep most of us out of the Aderhold parking lot.

This page demonstrates some linkages used to produce straight-line motion. In general, proofs are not provided. These explorations were mainly used to try and develop some expertise in using Geometer's Sketchpad, however, some mathematical investigations are included. A list of references is provided.

Lent (1961) states that

"Guiding a point along a straight line presents very few problems today, since it is a simple matter to produce very accurate plane surfaces along which a member may slide. Before James Watt could build his steam engine, he had to design a linkage to guide a pin along a straight line path, since there were no machine tools in 1769 capable of producing straight metal slides of sufficient precision."  (p 15)

Pivot points A and B are fixed. The segments represent arms of fixed length, connected at A' and B'.
As AA' rotates about point A, the midpoint M of arm A'B' traces out approximately a straight line through part of its path.
Click anywhere on the figure to open Geometer's Sketchpad and explore how changing the lengths of the arms or the distance AB effects the path of point M.

As an interesting side note, if  A'B' = AB and AA' = BB' = AB/root 2, point M traces out a Lemniscate of Bernoulli as shown below.

The product MB * MA is a constant, so M lies on a lemniscate of Bernoulli. For a proof that the green curve is a lemniscate of Bernoulli, click here to open a Microsoft Word document.
Click on the figure to open GSP for an animation AND a limitation of GSP!! You'll see why I include only half of the lemniscate in the figure above.

There are several possible configurations of  Watt's linkages as shown here.
None of them overcome the limitation of GSP, but by reflecting some of the objects in the linkage, you can "trick" GSP into drawing only the lemniscate.

Another straight line linkage is attributed to Roberts (1789 - 1864). It consists of two equal lengths crank arms AC and BD with a connecting rod CD as shown below. Triangle CPD is isosceles and the path of point P approximates a straight line as AC rotates about an arc of circle A. Click on the figure to explore this linkage with GSP.

Click the figure to open GSP and explore this linkage.

Points A and B are fixed pivot points. The lengths are as shown.
Point P traces a nearly straight path as AC rotates about point A.

This was the first linkage proved to draw a true straight line throughtout its range of motion.
Click the figure to open GSP and see the animation of this linkage.

EK and EL are equal length arms and FLF'K is a rhombus.
As F follows a curve, F' traces the inverse of the curve. With F constrained to a circle, F' traces a straight line.

The epicyclic straight line mechanism.
While not based on linkages, a line segment is produced by this device. A circle rolls around the inside of a second circle twice the diameter of the first. A point on the smaller circle traces out the diameter of the larger circle. Click here to see a GSP sketch and animation of this mechanism.

Using dynamic geometry software to reproduce mechanical devices or to recreate classical methods for drawing common mathematical curves is a rich area for exploration. For example, how do you use geometry to draw a limaçon? Click here to see one way.

References

The books referenced below are from the University of Georgia library, however much of the same information is available on the World Wide Web.

Dijksman, E. A. (1976). Motion geometry of mechanisms. London: Cambridge University Press.

Huckert, J. (1958). Analytical kenematics of plane motion mechanisms. New York: The Macmillan Company.

Hunt, K. H. (1978). Kinematic geometry of mechanisms. Oxford: Clarendon Press.

Lent, D. (1961). Analysis and design of mechanisms. (p 15). Engelwood Cliffs, NJ: Prentice-Hall, Inc.

McCarthy, J. M. (2000). Geometric design of linkages. New York: Springer.

Schwamb, P., Merrill, A. L., James, W. H. (1948). Elements of Mechanism. (Rev. ed.). New York: John Wiley and Sons, Inc.

Weisstein, E. W. (2001). Eric Weisstein's world of mathematics. Retrieved December 22, 2001 from
http://mathworld.wolfram.com/.

Wells, D. (1991). The Penguin dictionary of curious and interesting geometry. London: Penguin Books.