Using Outer Product for Collision Detection

Introduction

Collision detection is a fundamental problem in computer graphics, physics simulations, and game development. Efficient algorithms for detecting collisions can significantly improve performance and responsiveness in real-time applications. This article explores the use of the outer product as a mathematical tool for enhancing collision detection algorithms.

Understanding the Outer Product

The outer product is a mathematical operation that takes two vectors and produces a matrix. If you have two vectors A and B in a three-dimensional space, the outer product is denoted as:

$$A \otimes B$$

This results in a matrix where each element is computed as $A_i \cdot B_j$.

Mathematical Representation

For vectors:

$$A = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}, \quad B = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}$$

The outer product results in:

$$\begin{pmatrix} a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & a_3b_3 \end{pmatrix}$$

Code Example

Here’s how you can implement the outer product in JavaScript:

function outerProduct(A, B) {
  const result = [];
  for (let i = 0; i < A.length; i++) {
    result[i] = [];
    for (let j = 0; j < B.length; j++) {
      result[i][j] = A[i] * B[j];
    }
  }
  return result;
}