Intersection of Line Segments in a Discrete Domain

The intersection of line segments has already been studied by several authors. Several problems occurs when using floating point representation, in which number comparison is performed using a tolerance value. One of the main problems is the non-transitive property, as it may lead to incorrect results. Thus, fixed precision is preferred because it solves several problems. However, fixed precision adoption has some complications. One of the issues is relates to the intersection determination between two or more line segments. Particularly, in this research, it is shown that current methods to determine intersection between line segments with fixed precision are not commutative; i.e., depending on which order the line segments are processed a different result may be obtained. The line segment partition problem must be solved beforehand. In order to approach these problems, two line segment representations are considered: geometric and set representation. To the best of our knowledge, this is the first work which discusses the non commutative problem of line segment intersection with finite precision.