Peaucellier's Linkage

Part 1. Construct the Linkage

1. Start with three segments off to the side which determine your three lengths.

2. Use Translation by Fixed Cartesian on the endpoints of one of the segments you just constructed to get U and T.

3. You can now construct P. The only requirement is that P is a specified distance from U. How can you construct it?

4. How do you determine the placement of R and S? Remember that they are both a specified distance from T and a specified distance from P. Construct them.

5. To determine the placement of Q, what do you know about a polygon with four sides of equal length? Mathematically, you could construct Q using either intersections of parallel lines or intersections of circles. However, for technical reasons, intersections of parallel lines works better. Construct Q.

6. After completing your sketch, move P while you trace P and Q to form conjectures as to the shapes their traces lie on.

7. Change the lengths of the three segment and move P while you trace P and Q again. Does your conjecture remain true?

By now you have formulated a conjecture for the shape traced by Q as P moves. The next question is: Why do P and Q behave as they do? In the next section, you will learn about inversion. In part 3, you apply this knowledge to your linkage to understand why P and Q move as they do.

The Geometry Center Home Page

Author: Evelyn Sander
Comments to: webmaster@www.geom.uiuc.edu
Created: Jun 09 1996 --- Last modified: Jun 11 1996