Non-Midpoint Segment Splitting in Ruppert's Delaunay Triangulation Refinement Algorithm Roughly speaking, in Ruppert's Delaunay Triangulation refinement algorithm, so called encroached edges are split until no more encroached edges remain.  

The algorithm specifies splitting the edges at their midpoint (except in the case of small input angles where concentric circular shells are suggested. This question is unrelated to these cases.) 

In certain domains, given a segment, there are points on the segment that I would prefer to split on that are not necessarily the midpoints (unrelated to the concentric shell trick). These points are chosen based on some domain specific underlying data (considerations beyond the graph structure the algorithm is aware of). 


What are the implications of splitting on non-midpoints?   
What needs to be taken into consideration when selecting among several non-midpoint candidates?   
Does splitting on non-midpoints affect any of the convergence properties
of the algorithm?


Another way to ask this is: Are there split points that are better (by some interesting measures) than the a-priori selected midpoints?
Alper Ungor did some nice work related to this issues of how do Delaunay refinement with alternative points. See his paper: 

A. Ungor. Off-centers: A new type of Steiner points for computing size-optimal guaranteed-quality Delaunay triangulations. Computational
Geometry: Theory and Applications (CGTA), 42(2): 109--118, 2009.