Mechanism for describing an algebraic curve

This question has been central to kinematics for several centuries, and its origins can be traced back to Watt, with the drawing of a straight line. It's important to understand the theoretical and practical implications. There's a big difference between generating a curve and reproducing one.

Let's illustrate this with the circle:

• by tracing the edge of a coin with a pencil, we draw a circle that is the reproduction of an existing circle;

• with one instrument, the compass, we can generate any circle of given radius, i.e. of given equation.

For the right, the answer is much more difficult:

• the common drawing of the line with the ruler is the reproduction of an existing line;

• ...