3D Math Primer for Game Programmers

Introduction

In computer graphics, we use transformation matrices to express a position in space (translation) as well as its orientation in space (rotation). Optionally, a single transformation matrix can also be used to express the scale or “shear” of an object. We can think of this transformation matrix as a “basis space” where if you multiply a vector or a point (or even another matrix) by a transformation matrix you “transform” that vector, point or matrix into the space represented by that matrix.

In this article, I will not discuss the details of transformation matrices. For a detailed description of transformation matrices, you can refer to my previous article titled Matrices.

In this article, I want to discuss an alternative method of describing the orientation of an object (rotation) in space using quaternions.

The concept of quaterinions was realized by the Irish mathematician Sir William Rowan Hamilton on Monday October 16th 1843 in Dublin, Ireland. Hamilton was on his way to theRoyal Irish Academy with his wife and as he was passing over the Royal Canal on theBrougham Bridge he made a dramatic realization that he immediately carved into the stone of the bridge.