In this paper two versions of the incremental algorithm for computing regular triangulations in the plane are proposed. Both algorithms have the advantage of being very simple to implement because they do not require any complex data structure than a queue or a list (except for the self triangulation). The algorithms and a TCL/TK interface that allows direct user interaction have been implemented. Results show that, even thought the algorithms have a worst case time comlexity of $O(n^2)$, the running expected time for not very large sets of points is significantly shorter.