The amazing book, 1000 PlayThinks,
by Ivan Moscovich (order it from
Amazon.com),
has this interesting question:Return to my Puzzle pages
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© Copyright 2002, Jim Loy
The amazing book, 1000 PlayThinks,
by Ivan Moscovich (order it from
Amazon.com),
has this interesting question:
Moving Along Circles: Imagine a straight linkage as shown here, with each end constrained to one of two intersecting circles. Can you puzzle out the path traced by the middle dot of one linkage as it moves through one full circle?
I have drawn something like his solution, above left. This is normally called a three bar linkage (draw bars from the ends of the bar shown to the centers of the circles). It is an interesting looking curve. Well, that is not the only curve possible. As we move these circles closer together or farther apart, we get the following curves (or "loci," plural of "locus" or path of a point).
In the next two, the circles don't intersect:
This is another puzzle from the same book. Instead of
a bar between the two intersecting circles (which seem to be of about equal
size in his diagram), he has a triangle (seemingly isosceles), with two
vertices sliding on the two circles. We are to deduce the path of the third
vertex. He gets a near-circle. Later in the book, with a smaller triangle, he
gets something like the picture on the left.
I have done some experimenting. On the right is one with a bigger triangle. Below left, I very nearly got his circle (with a couple of ear flaps). And below right, I moved the circles closer together.
Below we have a larger triangle, and different sized circles:
Addendum:
Here is the same kind of thing, drawn as a three bar linkage. I have extended one bar to subtly change the shape of the curve:
More complicated linkages are possible. It has been shown that a finite piece of any algebraic curve can be drawn with some kind of linkage. Here is a curve drawn with a three bar linkage, with an added bar that is hinged at one point and slides through a second point:
I've animated it at Butterfly (requires Java).
I have drawn these curves using Cinderella and Paint Shop Pro.