Let G be a Lie group acting smoothly on the plane. Then two smooth curves C and C’ are G-equivalent if there exists some g in G such that gCC’. Can we answer the question, when are two curves G-equivalent? What can we additionally say if we restrict our attention to algebraic curves? In this talk we will first introduce the signature map and explain how it can help identify when two smooth curves are G-equivalent. Then we will investigate the signature map on algebraic curves and what additional information we can glean using Bezout’s and Bertini’s Theorem. The talk will be accessible and will review much of the background information. This is joint work with Drs. Irina Kogan and Cynthia Vinzant.