  We characterise the quotient surface graphs arising from symmetric contact
systems of line segments in the plane and also from symmetric pointed
pseudotriangulations in the case where the group of symmetries is generated by
a translation or a rotation of finite order. These results generalise well
known results of Thomassen, in the case of line segments, and of Streinu and
Haas et al., in the case of pseudotriangulations. Our main tool is a new
inductive characterisation of the appropriate classes of surface graphs. We
also discuss some consequences of our results in the area of geometric rigidity
theory.
