Creating Linkages

One of the most interesting geometric constructions is that of a linkage of fixed bars that travels a certain path. The construction of linkages is a part of geometry that has a rich and interesting history.  As a first example of a linkage let's look at how one would construct a series of three bars of fixed lengths that are attached at the endpoints.

We start with a set of three segments that will serve as the fixed lengths of our three bars.  Construct three segments a, b, and c in the Canvas.

3bar 1

Now, we need to create three new segments that are attached to each other and have these three lengths.  To do this, we note that if a segment of fixed length is attached to another of fixed length, then the attachment point lies on a circle.  Thus,  to create the first segment and an attachment point for the second segment, we create a point A, a circle with center A and radius the length of segment a, and then attach a point B onto this circle.

3Bar 2

Now, at B we will want to again create a circle of radius the length of segment b for the next segment.  However, we also want this next segment to be attached to a third segment having length equal to the length of c.  To do this, let's create another point C and a circle centered at C of radius equal to the length of segment c.

3Bar 3

In the final step of this construction, we will want to have a segment with endpoint  B of length of segment b, and this segment attached to a segment with endpoint at C of length of segment c.  An easy way to do this is to create a circle with center B and radius equal to the length of segment b and then find the intersection of this circle with the circle centered at point C.  Once this intersection is determined we create the three bars.

3Bar 4

Note that when we create the third bar we need to make a choice of which of the two circle-circle intersection points to use.  If we now grab point B and move it we will see the linkage moving much as a physical linkage of bars would move.

3Bar 5

However, when point B moves too far around circle A and the intersections disappear, then the segments based on these intersections will also disappear.

3Bar 6