  We apply numerical algebraic geometry to the invariant-theoretic problem of
detecting symmetries between two plane algebraic curves. We describe an
efficient equality test which determines, with "probability-one", whether or
not two rational maps have the same image up to Zariski closure. The
application to invariant theory is based on the construction of suitable
signature maps associated to a group acting linearly on the respective curves.
We consider two versions of this construction: differential and joint signature
maps. In our examples and computational experiments, we focus on the complex
Euclidean group, and introduce an algebraic joint signature that we prove
determines equivalences of curves under this action and the size of a curve's
symmetry group. We demonstrate that the test is efficient and use it to
empirically compare the sensitivity of differential and joint signatures to
different types of noise.
