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This paper considers the problem of designing a surface separating two homogenousmedia I and II that refracts a collimated beam of rays from a domain Rn into a target destination T() Rn+1. We find conditions on the transformationT such that a refracting surface exists locally. For example, given a planeimage F surrounded by medium I and an image F0 in 3d-space surrounded bymedium II, we find conditions so that there is a surface separating the media thatrefracts F into F0 locally. We also consider in Section 3 a similar question when therays emanate from a point source. The main result is Theorem 2.1 and examplesof illustration are given in Subsection 2.2. Our method also gives similar resultsfor reflection. With the same method we also show in Subsection 2.3 that givena plane curve and a space curve 􀀀 satisfying (2.17), there exists a surface thatrefracts into 􀀀. All problems in the paper are solved by studying the systems offirst order pdes (2.2) and (3.10).