  Conformal transformation optics provides a simple scheme for manipulating
light rays with inhomogeneous isotropic dielectrics. However, there is usually
discontinuity for refractive index profile at branch cuts of different virtual
Riemann sheets, hence compromising the functionalities. To deal with that, we
present a special method for conformal transformation optics based on the
concept of geodesic lens. The requirement is a continuous refractive index
profile of dielectrics, which shows almost perfect performance of designed
devices. We demonstrate such a proposal by achieving conformal transparency and
reflection. We can further achieve conformal invisible cloaks by two techniques
with perfect electromagnetic conductors. The geodesic concept may also find
applications in other waves that obey the Helmholtz equation in two dimensions.
