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

Flying maneuvers

Any object in 3D space can have transformations applied to it. It’s no different for the camera. In this section, we will explore how the same mathematics can be applied to the object to move and reorient it.

In Chapter 12, Mastering Affine Transformations, we discussed the three rotations that can take place in 3D space; namely, pitching, yawing, and rolling. These correspond with rotations around the x, y, and z axes respectively.

The 4 x 4 matrix that will perform a pitch around the x axis is:

The matrix that will perform a yaw around the y axis is:

The matrix to perform a roll around the z axis is:

We can use these matrices to rotate the camera and thus the view space. To do this, we multiply the view matrix by one or more of these.

The way the camera, or any other object transformed with these matrices, rotations can be likened to the movement of an aircraft as illustrated in Figure 15.1. In Figure 15.1 (a), the rotational...