I'm trying to optimize some reflective properties of curves of the form: $a_1x^n+a_2x^{n-1}+a_3x^{n-2} + ... + a_n + b_1y^n+b_2y^{n-1}+b_3y^{n-2} + ... + b_n = 0$

which is basically the curve that you get when you sum two polynomials of same degree in different variables:

$f(x) + g(y) = 0$

Anyways, I was wondering what would be a good way to do crossover on two such curves? I tried averaging the curves and then mutating them, but the problem is that the entire population quickly becomes homogeneous and the fitness starts to drop. Another typical method I tried is taking a cutoff point somewhere on the expansion above and mixing the left and right part from both parents.

Of course there are many ways to do a similar process. I could order the above expansion differently and then do a cutoff. I could seperate Xs and Ys and do two seperate cutoffs. Etc. The question is: in the context of algebraic curves, which method of generating offspring would be a good option considering I want to optimize some property of the curve (in this case I want it to have some reflective properties)?


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