Problem Solving from First Principles with JavaScript: A Case Study

Core Problem-Solving Philosophy

• 💡 The course emphasizes building solutions from first principles using simple reasoning and foundational math, rather than adapting existing methods.
• 🧠 The goal is to develop a powerful thought process for tackling complex problems.
• 🛠️ Algorithms are implemented from scratch and explained step-by-step with simple language and visuals.

Case Study: Vision-Based Positioning System

• 🎯 The central problem is reconstructing drawings from videos captured by a pen with a camera, using a vision-based positioning system.
• 🧩 This complex problem is broken down into smaller, manageable sub-problems, with techniques taught for identifying, prioritizing, and working on interconnected issues independently.
• 📐 Initial approaches involve analyzing apparent size differences of objects (balls) due to perspective to estimate distances.

Algorithmic Approaches and Concepts

• 📏 Trilateration is introduced as a method to determine position using estimated distances from known points, similar to GPS systems.
• 🌐 The concept of camera distortion is explored, explaining how lenses and flat projection planes can stretch images, especially at the edges.
• 🧮 Local search and optimization techniques are presented as general methods for solving complex problems, including finding camera parameters and estimating positions by minimizing error or maximizing a score.
• 🔄 Simulation using 3JS is used to create a controlled environment for testing algorithms and understanding camera behavior, including modeling the pen, camera, and ball movements.

Technical Implementation and Refactoring

• 💻 JavaScript is used for its ease of use and visualization capabilities, with code structured into multiple files for better management (parameters, math, image processing, estimation, simulation, etc.).
• 📈 Concepts like image data manipulation, canvas rendering, vector operations, and 3D transformations are implemented from scratch.
• 🗄️ The importance of refactoring code to maintain clarity and manageability as complexity increases is highlighted, with examples of separating concerns into different modules.

Key Mathematical and Geometric Concepts

• 📐 Pythagorean theorem and Euclidean distance are foundational for calculating distances between points.
• 🔄 Trigonometry (sine, cosine, tangent) is essential for understanding angles, rotations, and projections.
• 🌐 Map projections (azimuthal equidistant, Lambert azimuthal equal-area) are discussed in the context of correcting image distortions.
• 🎛️ Local search and gradient descent are presented as powerful optimization techniques applicable to various problems, from finding camera positions to solving the Traveling Salesperson Problem (TSP).