This paper tackles the problem of the computation of a planar polygonal curve from a digital planar curve, such that the digital data can be exactly retrieved from the polygonal curve. The proposed transformation also provides an analytical modelling of a digital plane segment as a discrete polygon composed of a face, edges and vertices. A dual space representation of lines and planes is used to ensure that the computed curve remains inside the digital curve, and this tool enables to define a very efficient algorithm. Applied on the digital plane segments resulting from the decomposition of a digital surface, this algorithm provides a set of polygons modelling exactly the digital surface.