The observation of the broad relativistic iron Kα line in the active galactic nucleus MCG-6-30-15 stimulated the development of numerical codes that correctly take into account all relativistic effects near the vicinity of a black hole, in order to model such lines. They are nowadays considered to be born in an accretion disc around a supermassive black hole. The properties of iron lines lead directly to the observable properties of accretion discs and of black holes as well.
Various numerical approaches have been suggested and used to calculate the profiles of emission lines: the historical method using the transfer function, direct integration of the trajectory of the photons, and elliptic integrals. We present new, fast and accurate methods that use integrated geodesic equations in terms of Jacobian elliptic functions, in both the Schwarzschild and the Kerr metrics, to produce accretion disc images and line profiles. Two different codes were designed in order to make maximal use of the symmetries of the Schwarzschild and Kerr solutions. In the Schwarzschild case, one can analytically map any point of the disc – even a warped one – into the image plane, which makes the calculations accurate and fast. In the Kerr case, the analytic form of the integrated geodesic equations is not as simple, so that some numerical interpolations are required in calculating line profiles, and the case of warped discs cannot be treated as easily. Both codes are able to calculate lines produced by material extending below the marginally stable orbit and can handle various emissivity and illumination laws.