In differential geometry we have the fundamental theorem of curves for two curves given parametrically:
If two plane curves, parametrized by their arc length, have the same curvature (as a function of the parameter), then the curves are congruent - there is an isometry of the plane that takes one curve to the other.
This theorem gives an (almost) effective procedure for determining whether two curves, given parametrically by functions which are computable, are congruent. Calculating curvature involves only derivatives and so is always tractable. So the only difficulty applying this method is that we need to parameterize both curves in terms of their arc length, which involves an integral. But, if we can somehow solve integrals, then we have a method for always determining whether two parametric plane curves are congruent. Generalizations of the above theorem also exist for space curves and for surfaces.
Algebraic curves however are also interesting, and they can rarely be parametrized, so we can't apply the above method. Curvature formulas exist for algebraic curves, but they also give the curvature implicitly (as a function of both x and y), so again, we can't apply the above theorem. So, is there a method to determine whether two algebraic plane curves are related by a euclidean isometry? I'm also fine with "partial" methods that depend on our ability to solve another common problem, such as solving integrals or finding roots of polynomials (although if somehow a complete method exists, that's obviously preferable). If such a method exists, a generalization for surfaces or space curves is also welcome.