# How do I draw an elliptic curve?

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?

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Computers...... –  Graphth Apr 21 '11 at 14:57
I wonder why people are upvoting this facetious answer.. –  quanta Apr 21 '11 at 15:38
Elliptic curves such as $y^2=x^3-x$ have two components. Could a device plot both components? –  lhf Apr 21 '11 at 17:28

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.

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That's quite neat but am I right in thinking it only lets us construct a single point at a time rather than tracing out part of the curve? –  quanta Apr 21 '11 at 17:22
Yes. This would be quite tedious. Though, there may be some way of streamlining it. The constructions have so many similarities. I've added some images to show this. –  futurebird Apr 21 '11 at 17:26

This thesis looks like it may be of considerable interest.

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