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

Combining transformation matrices for complex maneuvers

As with the OpenGL order of transformations, which we discussed in Chapter 12, Mastering Affine Transformations, when combining these homogeneous representation matrices to produce compound movements involving translation, scaling, and rotation, the matrices are presented in reverse order. For example, to transform a point by (3, 4, 5), rotate it around the X-axis by 45 degrees, and then scale it by 0.3 in all directions; the matrix multiplication is as follows:

Note how the translation matrix of the first operation is placed on the right and the scaling matrix on the left. To multiply this out, we begin by multiplying the last two matrices (the translation and rotation) to get the following:

Then, we complete the multiplication with the remaining two matrices, which results in the following:

Although learning to calculate these operations by hand is a great skill to have and will help embed your understanding...