Peaucellier Linkage

Peaucellier's linkage is designed to transform circular motion into linear motion.  It is based on three fixed lengths a, b, and c.  To construct the linkage we must have a fixed point and a point (center) that is a constant distance (equal to the length of b) away from the fixed point. An easy way to do this is to construct a circle at the center point of radius b and attach the fixed point to it, as shown.  Then, we attach a movable point (mover) to the circle.  Next, we construct two points that are equal distance (length of c) from the mover point and that are also equal distant (distance of a) from the fixed point, as shown.  Then, we create two segments (in blue) from the mover to these two new points. Finally, we create a rhombus on these two new segments by reflecting them across a segment between the new points.  Grab the mover point and drag it back and forth. What does the linePt trace out. [Big hint: consider how it was named!]

    Place a trace on the point labeled "linePt" to convince yourself that your conjecture is true.
    What happens if we try this same construction in the hyperbolic plane?

If we trace the point "Pt" in this case we get an interesting curve. But, it is not a line, as the black line connecting the ends of the trace does not coincide with the trace.  Also, it does not appear to be an arc of a circle.  The nature of this traced curve appears to be an open question.