Locus of points traced by the Chebyshev lambda linkage

Question

I was looking around on the interwebs when I found a GIF of the locus of points traced by this mechanism called the Chebyshev lambda linkage, and I was curious if there is an equation that describes the baguette-shaped locus of points traced by it. Can you give me some pointers on how to go about deriving such a formula?

Here's that GIF I mentioned:

Answer 1

The caption below the GIF as shown on the Wikipedia article gives the dimensions:

Dimensions:
Cyan Link $=a$
Green Link $=2.5a$
Yellow Link $=2.5a+2.5a$
Horizontal Distance between Ground Joints $=2a$

Fix $a=\frac12$ and let the two fixed joints lie at $(-1,0)$ (left) and $(0,0)$ (right). Then if the angle of the cyan link is $t$ so the cyan/yellow joint $A$ is at $(-1+\frac12\cos t,\frac12\sin t)$:

• The distance from $A$ to the right fixed joint/origin is $a=\sqrt{5/4-\cos t}$
• The distance from the origin to the tip joint (that traces the curve) is by the Pythagorean theorem $b=\sqrt{5+\cos t}$
• The tip joint's position vector is $A$'s position vector rotated clockwise $90^\circ$ and scaled to length $b$: $$\left(\frac12\sin t,1-\frac12\cos t\right)\frac ba=(\sin t,2-\cos t)\sqrt{\frac{5+\cos t}{5-4\cos t}}$$ This is a parametric equation for the curve.