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

Using the parametric form of lines

While the line equation given in the previous section is something most people are familiar with from high school mathematics, it’s not particularly useful in graphics when you want to manipulate objects or work out intersections, animations, and collisions. Therefore, we tend to use the parametric form. The parametric form of an equation, rather than using x and y to calculate positions, uses time, represented by t. This might sound confusing at first but bear with me while I explain.

Consider Figure 10.3 (a). Notice how a line segment can be represented by two points and a vector going between them:

Figure 10.3: A line segment with a vector between the start and end points

The calculation for v is as follows:

v = b – a

We can also express it like so:

b = a + v

This tells us that if we start at point a and travel along the whole length of v, we will end up at b. Where would you be if you only...