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

The Naïve Way: Drawing a line with brute force

So, you have an equation to draw a line and want to plot it on the screen pixel by pixel. The naïve approach would be to loop through a series of x values, calculate what the y value should be at that x location given the equation, and then plot x, y as a colored pixel. When you do, a very unfortunate issue will arise.

An issue occurs because pixel locations are represented as whole numbers; however, points on a straight line can in fact be floating points. Consider the line in Figure 3.1, showing a grid of pixel locations and a line drawn through them:

Figure 3.1: A grid of pixels and a line

If you were to draw every pixel the line touched (shown by the shaded squares), it wouldn’t give a very accurate presentation of the line. In addition, the real location of any point on the line could calculate to a floating-point value. When this occurs, the value is rounded to an integer to find the closest...