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

Understanding the difference between points and vectors

We can make sense of vectors by examining them in Cartesian coordinates. A vector in 2D is expressed as (x, y), in 3D as (x, y, z), and in 4D as (x, y, z, w).

Yes, I said four dimensions! At this stage, you are most likely looking at that “w” at the end of the expression and wondering where it came from. Don’t worry about it too much as its purpose will become clearer when we examine matrix multiplication.

In theory, a vector can be defined in any number of dimensions extending to infinity. They are used for complex mathematical calculations that can be found in applications relating to machine learning, astrophysics, financial analysis, and inverse kinematics, to name a few. However, in graphics, 2D, 3D, and 4D vectors are used.

Figure 9.1 illustrates a point and a vector in both 2D and 3D space. If you were to just look at the expressions for a vector, shown previously, you could be forgiven for...