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 around an arbitrary axis

A vector lying on the x axis that is represented by (1, 0) and rotated by results in the vector that will be the cosine of the angle and the sine of the angle ( as illustrated in Figure 16.1. Likewise, a vector sitting on the y axis represented by (0, 1), when rotated by the same angle, will result in a vector that, too, contains a combination of cosine and sine as (-.

Figure 16.1: Two-dimensional rotations

Do these values look familiar? They should because they are the values we’ve used in the rotation matrix for a rotation around the z axis in Chapter 15, Navigating the View Space. Rotating in 2D is essentially the same operation as rotating around the z axis; as you can imagine, the z axis added to Figure 16.1 coming out of the screen toward you, and thus rotations in this 2D space are, in fact, rotating around an unseen z axis.

All the rotations we’ve looked at thus far have been to rotate a vector or...