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

• CLRS.Chapter33.segmentIntersect_of_sharesEndpoint — every shared endpoint is accepted by the CLRS intersection predicate

Status: partial — six cross-product algebra theorems, orientation_spec, and the shared-endpoint intersection case are proved. The segmentIntersect and bboxIntersect definitions still need full soundness and completeness against an independent geometric-intersection specification.

Deferred Work

• 33.1 general correctness of the line-segment intersection predicate, beyond the proved shared-endpoint case

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

• 33.4 Closest-pair divide-and-conquer