Math & Geometry: Overview

Typical Business Scenarios

Mathematical algorithms are often the hidden engines behind visual and security systems:

• Security (RSA): How do we calculate $A^B \pmod M$ when $B$ is a 2048-bit number? (Binary Exponentiation).
• Game Development: Detecting collisions or calculating the “vision cone” of an enemy (Geometry / Convex Hull).
• Finance: Pricing complex derivatives by simulating millions of possible market futures (Monte Carlo).
• Data Layout: Calculating the optimal grid size or scheduling periodic tasks (GCD / LCM).

Selection Framework: How to Choose?

1. Are you dealing with huge numbers?
   • Yes: Use Binary Exponentiation for powers and Euclidean Algorithm for divisors.
2. Is the problem deterministic or probabilistic?
   • Deterministic: Use exact formulas (Geometry, Matrix Math).
   • Too complex for formulas: Use Monte Carlo Simulations to approximate the answer.
3. Are you processing 2D/3D shapes?
   • Boundary Finding: Use Convex Hull (Graham Scan).
   • Intersection: Use Line Sweep algorithms.

Quick Look at Common Algorithms

• 7.1 Binary Exponentiation: Calculating $x^n$ in $O(\log n)$ time. Essential for cryptography.
• 7.2 GCD (Euclidean Algorithm): The 2,000-year-old algorithm that is still the fastest way to find the Greatest Common Divisor.
• 7.3 Monte Carlo: Solving impossible integrals by throwing random darts at a board.
• 7.4 Convex Hull (Graham Scan): Wrapping a set of points in the tightest possible rubber band.

Selection Cheat Sheet

The “One-Sentence Mindset”

“Math algorithms are shortcuts. They use properties of numbers and space to skip billions of unnecessary calculation steps.”