Definition of a bend: ends of the line should always be bends, otherwise not all line vertices are covered by bends (definition elsewhere). Gentle inflection at the end of the bend: the article does not specify how many vertices should be included when calculating the end-of-bend inflection. We chose the iterative approach -- as long as the angle is "right" and the distance is (greedily) decreasing, keep going. Self-line crossing when cutting a bend: the self-line-crossing may happen after a few bends have been skipped. E.g. ends of A<->B cross the line, but "swallow" a few more in between: ,______ / \ |___A | \ | \ | B\ | __ \ | | | / \ / | | |___,---,___/A | / | \_________________| \ | \ | / / ----/ / / ,____/ / B| | If a bend with 180+ deg inflection is found, its line between inflection angles (AB in our examples) must be crossed with all the other bends to detect a possible line-crossing. This is O(N*M), where N is the total number of line segments, and M is the number of qualifying bends. In other words, can be very computationally expensive. This may be slightly computationally simplified: if other bend's endpoints (A' and B') are in different sub-planes as divided by AB, then the crossing exists, and more expensive st_split can be used.