Chapter 33 — Computational Geometry

Chapter 33 of CLRS covers computational-geometry algorithms: line-segment properties, sweep-line segment intersection, convex-hull construction, and closest-pair finding.

This chapter currently formalizes Section 33.1 with fully proved cross-product and orientation theorems.

Sections

33.1 Line-Segment Properties

• CLRS.Chapter33.Point, CLRS.Chapter33.Vector — 2D point and vector types

• CLRS.Chapter33.cross — 2D cross product with antisymmetry, bilinearity, and additivity lemmas

• CLRS.Chapter33.Orientation — inductive Counterclockwise | Clockwise | Collinear

• CLRS.Chapter33.Segment — line-segment structure with bounding-box and intersection predicates

Status: partial — six cross-product algebra theorems and orientation_spec are proved. The segmentIntersect, bboxIntersect, and sharesEndpoint definitions still need correctness theorems, in particular soundness and completeness against an independent geometric-intersection specification covering endpoint and collinear cases.

Deferred Work

• 33.1 correctness of the line-segment intersection predicate

• 33.2–33.3 Sweep-line segment intersection and Graham-scan convex hull

• 33.4 Closest-pair divide-and-conquer