Chapter 6 Sampling and the Moments Technique

x " I've never touched the hard stuff, only smoked grass a few times with the boys to be polite, and that's all, though ten is the age when the big guys come around teaching you all sorts to things. But happiness doesn't mean much to me, I still think life is better. Happiness is a mean son of a bitch and needs to be put in his place. Him and me aren't on the same team, and I'm cutting him dead. I've never gone in for politics, because somebody always stand to gain by it, but happiness is an even crummier racket, and their ought to be laws to put it out of business. " – – Momo, Emile Ajar 6.1 Vertical Decomposition σ Given a set S of n segments in the plane, and a subset R ⊆ S, let A | (R) denote the vertical decomposition of the plane, formed by the arrangement A (R) of the segments of R. This is the partition of the plane into interior disjoint vertical trapezoids formed by erecting vertical walls through each vertex of A | (R). Formally, a vertex of A | (R) is either an endpoint of a segment of R, or an intersection point of two of its segments. From each such vertex we shoot up (similarly, down) a vertical ray till it hits a segment of R, or it continues all the way to infinity. See figure on the right. Note, that a vertical trapezoid is defined by at most 4 segments: two segments defining its ceiling and floor, and two segments defining the two intersection points that induce the two vertical walls on its boundary. Of course, a vertical trapezoid might be degenerate, and thus defined by less segments (i.e., an unbounded vertical trapezoid, or a triangle). Vertical decomposition breaks the faces of the arrangement, that might be arbitrarily complicated , into entities (i.e., vertical trapezoids) of constant complexity. This make handling arrangements much easier computationally.