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

Defining lines, segments, and rays

What most people call a line is a line segment. A line segment is just a piece of a line. By true mathematical definition, a line continues infinitely. In Chapter 2, Let’s Start Drawing, we used the equation for a line to draw segments between mouse clicks in our project window. Recall that the following equation was used:

y = mx + c

Here, m is the gradient (or slope) and c is the y-intercept, as shown in Figure 10.1:

Figure 10.1: The gradient and y-intercept

The value of y can be calculated for infinite values of x and vice versa, making a line continuous. However, a line segment has a start and end, as illustrated in Figure 10.2:

Figure 10.2: Differing straight geometrics

To clarify the difference between these straight geometrics, let’s examine their properties:

  • A vector has a magnitude and direction, as we discovered in Chapter 9, Practicing Vector Essentials. It doesn...