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We propose a new method for the reconstruction of a reflecting (refracting) surface from a given source-target map defining the relationships between the directions of incident and reflected (refracted) rays. In the proposed method, the optical surface is represented as an envelope of a set of paraboloids (reflecting surface) or ellipsoids (refracting surface). This representation allows the problem of design of an optical surface to be reduced to the reconstruction of a function from its total differential. We illustrate the proposed approach by designing mirrors generating a far-field uniform illuminance in a square target. The calculation results show that the proposed method enables the generation of high-quality illuminance distributions even when the integrability condition is not satisfied.

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