Classification of maps sending lines into translates of a curve

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 phi: R2 -> R2 and continuous curves C subset of R2 such that phi(t + C) is a line for all t is an element of 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 omega(n4/3) unit distances among n points in the plane.(c) 2023 Elsevier Inc. All rights reserved.