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

Scaling points with x, y, and z

It might seem a strange proposition to scale a single point if you think about it, as a point has no size—it’s just a location in space. So, what happens if you try to scale it through the affine transformation of scaling? Well, scaling is an operation performed by the multiplication of each of the point coordinates. Take, for example, the point (2, 4, 6)—if this is scaled by 0.5 (in other words, halved), the resulting point is (1, 2, 3). In this case, what has happened to the point is that it has been moved.

The formal mathematics for scaling is:

P(x, y, z) = S x Q(x, y, z)

Here, the x, y, and z coordinates of the resulting point P are the point Q’s individual coordinates multiplied by S. Let’s consider again the cube from Figure 12.2. The result of multiplying each of the cube’s vertices by 0.5 will result in the cube shown in Figure 12.4:

Figure 12.4: A scaled cube

In this...