Cage coordinates are a powerful means to define 2D deformation fields from sparse control points. We introduce Conformal polynomial Coordinates for closed polyhedral cages, enabling segments to be transformed into polynomial curves of any order. Extending classical 2D Green coordinates, our coordinates result in conformal harmonic deformations that are cage-aware. We demonstrate the usefulness of our technique on a variety of 2D deformation scenarios where curves allow artists to perform intuitive deformations with few input parameters. Our method combines the texture preservation property of conformal deformations together with the expressiveness offered by Bezier controls.