Equation for a Vinyl curve

Question

This video seems to show an explicit map from the torus to $\mathbb{R}^2$. Does it factorize through the projection $\mathbb{R}^3 \to \mathbb{R}^2$? What is the equation of the curve?

Answer 1

These are coupler curves of a 6-bar mechanism driven by two cranks. The single crank driven 4-bar mechanism generates an Atlas of curves e.g., by Hrones and Nelson. They are derivable by 2D positioned vectors with link constraints.(Mechanical Engineering)