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

Rotating points around a pivot

Right now, you are probably feeling pretty comfortable with affine transformations. There’s no doubt that scaling and translation are simple concepts. But now, we move on to rotations. Hold on to your hat because the mathematics is about to go up a few notches in complexity.

Just as translation and scaling in 3D work with each of the x, y, and z axes, so too does rotation. An object can rotate around its x, y, or z axis. These rotations are illustrated in Figure 12.6:

Figure 12.6: A teapot rotated around each axis

What makes the calculations for each of these rotations more difficult than scaling and translation is that while the x, y, and z values applied in scaling and translation only affect their coordinate counterparts (for example, x affects x, y affects y, and likewise), with rotations, to rotate around one axis, the other two axes are involved. The following equations use what are called Euler angles. These...