C++ 3D数学基础：图形与游戏开发入门

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"C++ 3D Math Primer for Graphics and Game Development" 本书是Fletcher Dunn和Ian Parberry合著的《C++ 3D数学入门：面向图形与游戏开发》。这本书主要关注的是计算机图形学和游戏开发中的3D数学基础，涵盖了用于3D图形编程的关键数学概念。在计算机图形学和游戏开发中，3D数学扮演着至关重要的角色。它涉及到向量、矩阵、变换、几何体建模以及碰撞检测等多个方面。以下是一些关键知识点的概述： 1. **向量**（Vectors）：向量是表示方向和大小的量，常用来描述3D空间中的位置或运动。向量加法、减法、标量乘法和向量点乘（内积）以及向量叉乘（外积）是向量运算的基础。 2. **矩阵**（Matrices）：矩阵是一种二维数组，常用于表示线性变换，如平移、旋转、缩放等。在3D空间中，4x4的齐次坐标矩阵可以用来一次性完成多个变换。 3. **变换**（Transformations）：包括平移（Translation）、旋转（Rotation）、缩放（Scaling）和剪切（Shear）。这些变换通常通过矩阵来实现，它们可以组合以创建复杂的物体运动和视角变化。 4. **坐标系**（Coordinate Systems）：理解局部坐标系和世界坐标系的概念至关重要。物体有自己的局部坐标系，而所有物体都在一个统一的世界坐标系下交互。 5. **投影**（Projection）：为了将3D空间的物体显示到2D屏幕上，需要进行投影操作，常见的有透视投影（Perspective Projection）和正交投影（Orthographic Projection）。 6. **视椎体**（View Frustum）：定义了摄像机可以看到的3D空间区域，用于裁剪超出可视范围的对象。 7. **顶点着色器**（Vertex Shaders）和**片段着色器**（Fragment Shaders）：在现代图形硬件中，这些着色器用于计算物体表面的属性，如位置、颜色和纹理坐标，以及最终像素的颜色。 8. **光照模型**（Lighting Models）：如Phong光照模型，用于模拟光源对物体表面的影响，包括环境光、漫反射和镜面高光。 9. **几何体建模**（Geometry Modeling）：包括基本形状的构建，如立方体、球体和圆柱体，以及多边形建模技术，如三角化和平滑组。 10. **碰撞检测**（Collision Detection）：确保游戏对象能够正确地相互作用，包括点与几何体的碰撞、几何体之间的碰撞等。 11. **法线映射**（Normal Mapping）和**环境映射**（Environment Mapping）：提升表面细节和真实感的技术，通过存储额外的表面信息来模拟更复杂的光照效果。 12. **四元数**（Quaternions）：用于表示旋转，相比欧拉角，四元数能避免万向节死锁（Gimbal Lock）问题。学习这些概念不仅有助于理解3D图形的工作原理，也是编写高效游戏引擎和渲染库的基础。书中会详细解释这些概念，并提供实用的C++代码示例，帮助读者将理论应用于实践。

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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