图形与游戏开发3D数学基础指南

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《3D数学入门：图形与游戏开发指南》是由Fletcher Dunn和Ian Parberry合著的一本专为图形和游戏开发人员设计的教材。本书深入浅出地介绍了三维数学在计算机图形学和游戏制作中的核心概念，旨在帮助读者掌握必要的数学技能，以便在实际项目中实现流畅的3D视觉效果和交互体验。在本书中，作者首先从3D数学的基本概念开始，引导读者理解向量、矩阵、坐标系和空间几何的重要性。他们详细讲解了三维空间中的变换，如旋转、缩放和平移，这些都是构建3D场景和对象的关键。此外，书中涵盖了线性代数的基础，如向量空间、基变换和行列式，这些是理解3D模型变形和动画的基础。章节1的“引言”部分解释了为何在当今的图形和游戏开发中数学如此至关重要，它不仅影响着图形渲染的精度，还直接影响游戏性能和用户体验。随着游戏引擎的复杂性和实时计算需求的增长，对3D数学的理解成为开发者必备的核心能力。后续章节可能包括： - **投影和光照**：介绍了如何将三维物体转换到二维屏幕上的投影技术，以及如何模拟真实世界中的光照效果，如漫反射、镜面反射和折射。 - **纹理映射**：探讨如何使用数学来处理纹理贴图，使之无缝地贴合在3D模型表面，实现逼真的表面细节。 - **碰撞检测和物理模拟**：涉及三维空间中的碰撞算法，以及如何利用刚体动力学原理实现自然的物体运动和交互。 - **几何建模**：介绍基础的几何形状（如球体、立方体等）创建方法，以及更复杂的模型如NURBS曲面的数学原理。 - **动画和关键帧**：讨论如何通过关键帧和插值技术创建平滑的动画序列，以及使用数学工具如贝塞尔曲线来控制动画路径。 - **图形优化**：探讨如何利用数学优化算法减少计算负载，提升游戏性能，例如视口裁剪和近似计算。《3D Math Primer for Graphics and Game Development》是一本实用且全面的教程，无论你是初学者还是经验丰富的开发者，都可以从中找到深入学习和巩固3D数学知识的宝贵资源。通过阅读这本书，你将能够更好地理解和应用数学理论，为你的图形和游戏项目带来更高的品质和效率。

1.3 What You Should Know Before Reading

This Book

The theory part of this book assumes a prior knowledge of basic algebra and geometry, such as:

Manipulating algebraic expressions

Algebraic laws, such as the associative and distributive laws

Functions and variables

Basic 2D Euclidian geometry

In addition, some prior exposure to trigonometry is useful, but not required. A brief review of

some key mathematical concepts is included in Appendix A.

For the practice part, you need to understand some basics of programming in C++:

Program flow control constructs

Functions and parameters

Object-oriented programming and class design

No specific compiler or target platform is assumed. No “advanced” C++ language features are

used. The few language features that you may be unfamiliar with, such as operator overloading

and reference arguments, will be explained as they are needed.

1.4 Overview

Chapter 1 is the introduction, which you have almost finished reading. Hopefully, it has

explained for whom this book is written and why we think you should read the rest of it.

Chapter 2 explains the Cartesian coordinate system in 2D and 3D and discusses how the Car

tesian coordinate system is used to locate points in space.

Chapter 3 discusses examples of coordinate spaces and how they are nested in a hierarchy.

Chapter 4 introduces vectors and explains the geometric and mathematical interpretations of

vectors.

Chapter 5 discusses mathematical operations on vectors and explains the geometric interpre

tation of each operation.

Chapter 6 provides a usable C++ 3D vector class.

Chapter 7 introduces matrices from a mathematical and geometric perspective and shows

how matrices can be used to perform linear transformations.

Chapter 8 discusses different types of linear transformations and their corresponding matri

ces in detail.

Chapter 9 covers a few more interesting and useful properties of matrices.

Chapter 10 discusses different techniques for representing orientation and angular displace

ment in 3D.

Chapter 1: Introduction

from 1596 to 1650. Descartes is not just famous for inventing Cartesian mathematics, which at the

time was a stunning unification of algebra and geometry. He is also well known for taking a pretty

good stab at answering the question “How do I know something is true?” This question has kept

generations of philosophers happily employed and does not necessarily involve dead sheep

(which will disturbingly be a central feature of the next section), unless you really want it to. Des

cartes rejected the answers proposed by the ancient Greeks, which are ethos (roughly, “because I

told you so”), pathos (“because it would be nice”), and logos (“because it makes sense”), and set

about figuring it out for himself with a pencil and paper.

2.1 1D Mathematics

You’re reading this book because you want to know about 3D mathematics, so you’re probably

wondering why we’re bothering to talk about 1D math. Well, there are a couple of issues about

number systems and counting that we would like to clear up before we get to 3D.

Natural numbers, often called counting numbers, were invented millennia ago, probably to keep

track of dead sheep. The concept of “one sheep” came easily (see Figure 2.1), then “two sheep”

and “three sheep,” but people very quickly became convinced that this was too much work. They

gave up counting at some point and invariably began using “many sheep.” Different cultures gave

up at different points, depending on their threshold of boredom. Eventually, civilization expanded

to the point where we could afford to have people sitting around thinking about numbers instead of

doing more survival-oriented tasks, such as killing sheep and eating them. These savvy thinkers

immortalized the concept of zero (no sheep), and while they didn’t get around to naming all of the

natural numbers, they figured out various systems whereby we could name them if we really

wanted to, using digits such as “1”, “2”, etc. (or if you were Roman, “M”, “X”, “I,” etc.). Mathe

matics was born.

6 Chapter 2: The Cartesian Coordinate System

Figure 2.1: One dead sheep

Figure 2.2: A number line for the natural

numbers

The habit of lining sheep up in a row so that they can easily be counted leads to the concept of

a number line, that is, a line with the numbers marked off at regular intervals, as in Figure 2.2. This

line can, in principle, go on for as long as we wish, but to avoid boredom we have stopped at five

sheep and put on an arrowhead to let you know that the line can continue. Clear thinkers can visu

alize it going off to infinity, but historical purveyors of dead sheep probably gave this concept little

thought, outside of their dreams and fevered imaginings.

At some point in history it was probably realized that you can sometimes, if you are a particu

larly fast talker, sell a sheep that you don’t actually own, thus, simultaneously inventing the

important concepts of debt and negative numbers. Having sold this putative sheep, you would in

fact own “negative one” sheep. This would lead to the discovery of integers, which consist of the

natural numbers and their negative counterparts. The corresponding number line for integers is

shown in Figure 2.3.

The concept of poverty probably predated that of debt, leading to a growing number of people who

could afford to purchase only half a dead sheep, or perhaps only a quarter. This lead to a burgeon-

ing use of fractional numbers, consisting of one integer divided by another, such as 2/3 or 111/27.

Mathematicians called these rational numbers, and they fit in the number line in the obvious

places between the integers. At some point, people became lazy and invented decimal notation,

like writing “3.1415” instead of the longer and more tedious 31415/10000.

After a while, it was noticed that some numbers that appear to turn up in everyday life are not

expressible as rational numbers. The classic example is the ratio of the circumference of a circle to

its diameter, usually denoted as p (pronounced “pi”). These are the so-called real numbers, which

include rational numbers and numbers such as p that would, if expressed in decimal form, require

an infinite number of decimal places. The mathematics of real numbers is regarded by many to be

the most important area of mathematics, and since it is the basis for most forms of engineering, it

can be credited with creating much of modern civilization. The cool thing about real numbers is

that while rational numbers are countable (that is, placed into one-to-one correspondence with the

natural numbers), real numbers are uncountable. The study of natural numbers and integers is

called discrete mathematics, and the study of real numbers is called continuous mathematics.

The truth is, however, that real numbers are nothing more than a polite fiction. They are a rela

tively harmless delusion, as any reputable physicist will tell you. The universe seems to be not

only discrete, but also finite. If there are a finite amount of discrete things in the universe, as cur

rently appears to be the case, then it follows that we can only count to a certain fixed number.

Thereafter, we run out of things to count — not only do we run out of dead sheep, but toasters,

mechanics, and telephone sanitizers also. It follows that we can describe the universe using only

Chapter 2: The Cartesian Coordinate System

Figure 2.3: A

number line for

integers (note the

ghost sheep for

negative

numbers)

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