Abstract
We list four types of planar curves such that arrangements of their translates are (locally) combinatorially equivalent to an arrangement of lines. We find them by characterising diffeomorphisms ϕ:R2→R2 and continuous curves C⊂R2 such that ϕ(t+C) is a line for all t∈R2. There are exactly five maps satisfying (at least locally) this condition. Two of them define the same curve, so we have four different curves. These can be used to define norms giving constructions with Ω(n4/3) unit distances among n points in the plane.
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