How do I draw an elliptic curve?

Question

I can draw a circle using a compass.

I can draw an ellipse using two focal points and a loop of string.

I think that you can draw an arbitrary conic with a "generalized" compass for which the pencil can slide in and out as it is rotated.

What instruments and devices can draw elliptic curves?

Answer 1

I have an answer, for a limited case, and it is not pretty. Elliptic curves have the form:

$y^2= x^3 + ax +b$

Suppose $b=0$ and $a < 0$

$y^2= x(x^2 + a)$

$y= \sqrt{x}\sqrt{x^2 + a}$

You can construct $\sqrt{x}$ as shown here.

And, $\sqrt{x^2 + a}$ with $a < 0$ is the distance from (x, -a) to the origin.

We have two constructible lengths so just construct their product:

...and you have y.

Answer 2

This thesis looks like it may be of considerable interest.