Three bar linkages

This is a three bar linkage. Drag the gray points to modify the length of the bars.

To construct three bar linkages, one needs to take care of the two different intersection points of two circles. The interactive geometry tool provides three different versions of 'intersection point of two circles:

• Left intersection point. One of the two circles is the first and the other one is the second circle. The left intersection point of the two circles is now the one a your left when you look from the center of the first circle to the center of the second.
• Right intersection point.
• Alternating intersection point. This is the version will will use for the three bar linkages. First, the alternating intersection point is just the left intersection point. But if it happens that the left intersection point is not defined (because the circles do not intersect at all!), the alternating intersection point switches - from now on, it will be the right intersection point until it happens again that the circles do not intersect.

Why this alternating intersection point? As we already mentioned before, with this feature, one is able to construct three bar linkages. Drag the points M1 or M2 in the following construction to understand the concept of an alternating intersection point:

Now all we need to do to get a three bar linkage, is the following:

• As a starting point, we again take two circles C1 and C2 with centers M1 and M2, respectively.
• Then, we define the first bar by first taking a 'point on a line' P on C2 and then connecting R with M2.
• To define the second bar, the alternating intersection point will be involved. Draw a circle C3 (with the radius defined by the distance between 2 other points) around R and call its alternating intersection point with C1 S. The second bar is now the line between S and R.
• For the third bar, just connect S and M1 by a line.

If you now drag the point R, you will see how it works. To animate R automaticly, all you need to do is to chose in the context menu: mode -> movie, and then click on the point R.

But why does this yield to what we wanted to get? Why doesn't the point R run around the whole circle in one direction, but turns around at the right point on the circle?

This behaviour is controlled by the 'run-type' of the 'point on a line' R. Compare the two animations below to understand the difference between the two run-types.

Here is a listing of all the run-types we implemented:

• 0: run forwards
• 1: run backwards
• 2: run forwards until an object is undefined, then turn around, etc.
• 3: run backwards until an object is undefined, then turn around, etc.
• 4: run forwards from the start to the end, then turn around and run back to the start, etc.
• 5: run forwards from the end to the start, then turn around and run forwards to the end, etc.

Combining the three bar linkage with the feature of running points, we can construct figures like the following:

The red line is the locus of the red point when the blue one runs. Drag the green points to modify the figure and the blue one to see the red point running on the figure. Again, you can start a movie (context menu -> mode -> movie -> click on the blue point).

References

• The MacTutor History of Mathematics archive contains an index of famous curves and the same index with Java applets.
• The Cubic Surface Homepage, which is a project of the Algebraic Geometry Group at the university of Mainz as well, enables you to play interactively with cubic surfaces in 3-space.

© Oliver Labs, Algebraic Geometry Group, University of Mainz, Germany

Interactive Geometry Project, impressum