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

Shearing and reflections

The last of the affine transformations are shearing and reflections. These often don’t have a lot of time spent on them in graphics because they aren’t often used and aren’t part of a typical graphics API. However, for completeness, we will add them here as they generate objects that retain parallelism and ratios.

Shearing is a translation along one dimension. Formally, we can define a point to be sheared thus:

Px = Qx + sQy

Py = Qy

This is a two-dimensional case as it can easily be demonstrated in 2D. You will notice that the y coordinate undergoes no change, while the x coordinate is its old self plus a scaled value of the old y coordinate. It can, of course, also be the other way around, with x staying the same and y changing. The result of shearing is to make an object appear to lean over, as demonstrated in Figure 12.11. It’s a donut shape, shown as the original shape on the left and the sheared version on the right...