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

Introducing quaternions

The minimum number of values needed to represent rotations in 3D space is three. The most intuitive and long-applied method for defining rotations, as we’ve seen, is to use these values as the three angles of rotation around the x axis, the y axis, and the z axis. The values of these angles can range from 0 to 360 degrees or 0 to 2 PI radians.

Any object in 3D space can be rotated around these axes that represent either the world axes or the object’s own local access system. Formally, the angles around the world axes are called fixed angles, while the angles around an object’s local axis system are called Euler angles. However, often both sets of angles are referred to as Euler angles. We covered the mathematics to apply rotations around these three axes in Chapter 15, Navigating the View Space, in addition to investigating when these calculations break down and cause gimbal lock.

Quaternions were devised in 1843 by Irish mathematician...