How to plot an algebraic curve?

Question

Is there an algorithm to plot an algebraic curve $f(x, y)=0$ ?

If the degree of $f$ is 4 or less, then there is a formula for the solution, so I think I can plot the curve (I don't know there is a better algorithm or not). What about the general case?

Answer 1

Yes there are many ways to do this. Here is one particular way:

1. Guess one root.
2. Apply some numerical optimization to find your way towards one true root from your guess.
3. When close enough to a root, iterate the following:
   1. Take a small steps in the tangent direction.
   2. Take steps in the gradient direction.

If your surface is in many dimensions you may want to somehow systematically generate some kind of a local ON gradient-tangent basis.

One good thing about this approach is that you get to choose your guess freely, you can pick for example a point close to the center of the screen or whichever area you want to visualize.

Answer 2

If you use Wolfram Alpha, it is quite easy : you just as for the contour "line" where $f(x,y)=0$.

I added a small example here.