I am currently constructing a method to approximate implicitly given plane curves and surfaces, which are smooth and single-sheeted.

Now I have finished writing a Matlab function doing all the procedure in the case of curves and I have tested it on the example of the circle. Everything works just fine, but I am wondering if it will work on more complicated examples as well. The circle is too simple in the sense of it having symmetries and strong geometric properties such as constant curvature.

I know some slightly more complicated examples of curves, but all of them either have too simple geometric properties or are not smooth or single-sheeted. I haven't really looked for examples of surfaces yet, but there are some that come into my mind (e.g. monkey saddle) which at least have varying curvatures. What I really am looking for is some "random curve" with the stated properties (smooth, single-sheeted, implicitly given), preferably closed (for a nice visualization). In particular it should have no symmetries. Also it should not be a graph of a function itself (since then my method would be meaningless). I find it surprisingly hard to find such "random" examples online or in the literature.

I am wondering:

  1. Is there some method or algorithm (or even Matlab function) to construct such "random" implicit curves or surfaces?

  2. If not, are there any "nice" examples in the sense above (i.e. with the stated properties, but no such things as symmetries etc.)

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