Altering the optics in one section of a linear accelerator beam line will in general cause an alteration of the optics in all downstream sections. In circular accelerators, changing the optical properties of any beam-line element will have an impact on the optical functions throughout the whole machine. In many cases, however, it is desirable to change the optics in a certain beam-line section without disturbing any other parts of the machine. Such a local optics manipulation can be achieved by adjusting a number of additional corrector magnets that restore the initial optics after the manipulated section. In that case, the effect of the manipulation is confined in the region between the manipulated and the correcting beam-line elements. Introducing a manipulation continuously, while the machine is operating, therefore requires continuous correction functions to be applied to the correcting quadrupole magnets. In this paper, we present an approach to calculate such continuous correction functions for six quadrupole magnets by means of a homotopy method. Besides a detailed derivation of the method, we present its application to an algebraic example, as well as its demonstration at the seeding experiment sFLASH at the free-electron laser FLASH located at DESY in Hamburg.