Any countable (infinite or not) linear (total) ordering can be represented by displaying all its elements on an axis in increasing order. Such a representation can be generated using only geometrical constructions based on coloured line segment extensions and rules to handle segment intersections. After a bounded time, the construction segments disappear and only the representation remains. The process starts with finitely many segments, so that unbounded acceleration effects are used to generate infinitely many segments for the representation. There is no outside machinery nor operator: any needed computation has to be carried out through the drawing. After providing some illustrative examples with ad hoc constructions, we prove our main results. One rational signal machine (bounded to use only rational coordinates) can generate the representation of any decidable linear ordering (i.e. the order between two elements is decidable by a Turing machine). In the general case, there is a signal machine able to generate the representation of any countable linear ordering (encoded in a real number).