6.838, Computational Geometry; Fall 2001

Point Location

Sergi Elizalde & David Pritchard

Navigating the Map

Lemma 6.1: Each face in a trapezoidal map of a set S of line segments in general position has one or two vertical sides and exactly two non-vertical sides

For any trapezoid D, define

top(D) - the line segment (either of S or the bounding box) which bounds D from above

bottom(D) - the line segment (either of S or the bounding box) which bounds D from below

We can also define the left and right endpoints of D, leftp(D) and rightp(D), although there are several different cases to consider (we consider only leftp, the other case is symmetrical):

a) the left side is a point at which top and bottom meet

b, c) the left side is formed by a single vertical extension of a point

d) the left side is formed by both vertical extensions of a point

e) the left side is the edge of the bounding box, which occurs only for a single trapezoid

For every trapezoid D, top(D), bottom(D), leftp(D) and rightp(D) define the edges and vertices of that trapezoid. So we will store trapezoids as a 4-tuple of points/edges.