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

Enter Bresenham: The improved approach

Jack Bresenham developed this algorithm while working for IBM in 1962. In short, it plots a straight line but doesn’t leave pixels disconnected from each other. The basic premise is that when drawing a line pixel by pixel, each successive pixel must be at least a distance of 1 in either the x or y (or both) direction. If not, a gap will appear. While the naïve approach we’ve just used creates a plot where all the x values are a distance of 1 apart, the same isn’t always true for the y values. The only way to ensure all x and y values are a distance of 1 apart is to incrementally plot the line from point 1 to point 2, ensuring that either the x or y values are changing by a maximum of 1 with each loop.

Consider the close-up of a steep line being plotted in Figure 3.4:

Figure 3.4: A line being constructed pixel by pixel

The values for dx (change in x values) and dy (change in y values) represent...