The lambda version of Chebyshev's linkage
- Zoltán Kovács
Technically it may be easier to construct a so-called lambda form of a linkage, because it can be possible to draw the full curve by using one single movement. Also from the mechanical point of view, a lambda model can be easier to use. The following Lego model is a lambda version of a 4-bar linkage which is not exactly Chebyshev's one, but somewhat similar to it.
The pen refill should be inserted into the top part of the model inside the connector peg. Then the model should be hold firmly at the blue connector pegs on the top of the yellow bar while drawing the curve by moving the pen refill. Here are the steps how to build this model:
Such a model can be described by a few parameters. The red and black bars are connected in the 5th position of the red bar and the 1st position of the black bar. We will denote this by writing R-B=5-1. Similarly, the red and white bars are connected in the R-W=15-1 positions. There are also connections of the two black bars in positions B-B=11-7. Finally, the white and black bars are connected in the W-B=5-1 positions. Actually, we could also use a different pen position than 11 (on the black bar), but we will fix it now to P=11. (We will use the numbering "right" to "left" for the second black bar.)
- Create a GeoGebra file which can visualize the resulting curve. Which polynomial order do you expect?
- Find some nice sets of positions which result in good-looking curves both by experimenting with the model and your GeoGebra file. Please share your discoveries with your colleagues!
- Try to create the Chebyshev's linkage as a lambda model by using the diagrams from Wikipedia. You will want to mirror the diagram horizontally. The following parameters will give a useful model: R-B=5-6, R-W=9-2, B-B=11-6, W-B=4-1, P=11. The yellow bar will have no role in this construction because it is too short to hold your hands. Instead, it makes more sense to use the yellow bar to substitute the first black bar. The blue connector pens can help holding the model firmly.
You can find additional linkages in GeoGebra's Automated Reasoning Tools plotter benchmarking database including many other constructions. Note that most of them does not contain Sánchez's trick, so you may move the moving point into invalid positions as well.