Expert · Module 28 · Lesson 4 of 4
Computational Geometry: 2D Cross Products, Convex Hull, and Sweep-Line
Replace error-prone floating-point angles and slopes with exact 64-bit integer 2D Cross Products (CCW orientation), build the Convex Hull in O(N log N) using Andrew's Monotone Chain, and detect intersections with Sweep-Line.
By the end you can
- Compute exact 2D orientation (counter-clockwise, clockwise, collinear) using the 64-bit integer cross product (B - A) x (C - A) with zero division or.
- Construct the Lower and Upper Convex Hull of N points in O(N log N) time using Andrew's Monotone Chain algorithm
- Test line-segment intersection and closest-pair / rectangle union problems using orientation predicates and Sweep-Line
Builds on Quick sort and partitioning, Stacks and the LIFO rule, Order & Stack Patterns: Binary Search on Answer, Monotonic Stack/Queue, and Sweep-Line.
01 · A question first
Computing slopes `(y2 - y1) / (x2 - x1)` or `atan2` angles to test if three points turn left or right divides by zero on vertical lines and suffers IEEE-754 floating-point rounding bugs on nearly collinear points. How does one 64-bit integer subtraction-and-multiplication formula eliminate both bugs forever?
Make a guess before you read on.
02 · The idea
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What's inside
- Intuitive Explanation & Real-World Analogy
- Core Concepts & Visual Walkthrough
- Pseudocode
- Multi-Language Implementations (C++, Java, Python)
- Time & Auxiliary Space Complexity Derivation
- Edge Cases & Common Beginner Mistakes
- Alternative Approaches (Brute Force vs. Optimal)
- Pattern Recognition Guide & Three-Stage Practice Plan
03 · Practice
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