The online abstraction problem for euler diagrams

A Euler diagrams are an accessible and effective visualisation of data involving simple set-theoretic relationships. Efficient algorithms to quickly com- pute the abstract regions of an Euler diagram upon curve addition and removal have been developed, but a strict set of drawing conventions (called wellformed- ness conditions) were enforced, meaning that some abstract diagrams are not representable as concrete diagrams. We present a variation and extension of the methodology which enables region computations for Euler diagrams under the relaxation of several drawing conventions. We provide complexity analysis and compare with the previous methodology. The algorithms are presented for generic curves, allowing for specialisations such as utilising fixed geometric shapes for curves that often occur in applications.