Book Image

Mathematics for Game Programming and Computer Graphics

By : Penny de Byl
5 (1)
Book Image

Mathematics for Game Programming and Computer Graphics

5 (1)
By: Penny de Byl

Overview of this book

Mathematics is an essential skill when it comes to graphics and game development, particularly if you want to understand the generation of real-time computer graphics and the manipulation of objects and environments in a detailed way. Python, together with Pygame and PyOpenGL, provides you with the opportunity to explore these features under the hood, revealing how computers generate and manipulate 3D environments. Mathematics for Game Programming and Computer Graphics is an exhaustive guide to getting “back to the basics” of mathematics, using a series of problem-based, practical exercises to explore ideas around drawing graphic lines and shapes, applying vectors and vertices, constructing and rendering meshes, and working with vertex shaders. By leveraging Python, Pygame, and PyOpenGL, you’ll be able to create your own mathematics-based engine and API that will be used throughout to build applications. By the end of this graphics focussed book, you’ll have gained a thorough understanding of how essential mathematics is for creating, rendering, and manipulating 3D virtual environments and know the secrets behind today’s top graphics and game engines.
Table of Contents (26 chapters)
1
Part 1 – Essential Tools
9
Part 2 – Essential Trigonometry
14
Part 3 – Essential Transformations
20
Part 4 – Essential Rendering Techniques

Working with unit quaternions

As discussed, quaternions remove the limitations involved in compounding Euler-angle rotations. In this section, we will concentrate on reprogramming the camera in our project to pitch and roll with quaternions.

Before quaternions are multiplied, we must ensure they are unit quaternions. That means they will have a length of 1. If we go back to thinking of quaternion spacing being a sphere encompassing Euclidean space, then a quaternion represents a vector from the origin of both spaces to the surface of the sphere. In Figure 16.9, the vectors q1 and q2 represent these. The vector representing a quaternion is four-dimensional, with the coordinates storing the angle and axis as .

Of course, you must remember we can’t see these four dimensions in our 2D/3D sphere diagram, but they are there. In Figure 16.8, q3 represents an invalid quaternion in that it extends beyond the surface of the sphere. However, the rotation that it is meant to represent...