Unity 3D游戏开发基础数学指南

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"《Basic Math for Game Development with Unity 3D》是一本针对游戏开发初学者的数学基础指南，由 Kelvin Sung 和 Gregory Smith 共同撰写，并由 Clover Wai 负责插图和插画。该书由 Apress 出版，提供了电子版和纸质版，具有 ISBN 编号，内容涵盖了与 Unity 3D 游戏开发相关的数学基础知识。" 在游戏开发，尤其是使用 Unity 3D 这样的强大引擎时，掌握一定的数学知识是至关重要的。本书主要针对那些想要进入游戏行业的初学者，旨在帮助他们建立必要的数学概念，这些概念在创建游戏时会频繁使用。 1. **向量和矢量运算**：向量是游戏开发中的基本元素，用于表示方向和速度。书中可能介绍了向量的加法、减法、标量乘法和点积等操作，这些都是控制游戏对象运动的基础。 2. **矩阵与变换**：矩阵在图形学中用于处理坐标变换，如平移、旋转和缩放。学习矩阵运算能让你理解如何在3D空间中移动和操纵物体。 3. **几何与碰撞检测**：理解基本的几何形状（如点、线、面）以及它们之间的关系对于实现游戏中的碰撞检测至关重要。这有助于确定物体是否相互接触，从而触发游戏事件。 4. **三角函数**：正弦、余弦和正切在计算角度和弧度时非常有用，特别是在处理旋转和轨迹计算时。 5. **线性代数**：包括行列式、逆矩阵和特征值等概念，这些在解决复杂的游戏逻辑和物理模拟中扮演着重要角色。 6. **概率和统计**：在游戏设计中，随机性和概率是常见的元素，比如角色的技能触发概率、物品掉落率等，需要了解概率理论来实现这些功能。 7. **算法和数据结构**：虽然不是传统意义上的数学，但排序、搜索和图论等算法对于优化游戏性能和设计关卡逻辑至关重要。 8. **四元数**：相对于传统的欧几里得空间，四元数提供了一种更有效的方法来处理3D旋转，避免了万向节锁问题。通过阅读这本书，读者将能够理解并应用这些数学原理到实际的 Unity 3D 项目中，从而提升游戏的视觉效果、物理模拟和交互体验。此外，书中可能还包括实践案例和示例代码，以帮助读者将理论知识转化为实际技能。

K. Sung and G. Smith, Basic Math for Game Development with Unity 3D,

https://doi.org/10.1007/978-1-4842-5443-1_1

CHAPTER 1

Introduction and

Learning Environment

After completing this chapter, you will be able to

• Know the details of what this book is about

• Understand the style that this book uses to present

concepts

• Install Unity and an Integrated Development

Environment (IDE) for developing programming code

• Access the accompanying source code and run the

example projects

• Understand the Unity terminology used throughout

this book

• Begin to appreciate the intricate details of math for

game development

Introduction

When you think of math in a video game, you may picture health bars,

attack stats, experience points, and other game mechanics. You may not

consider the underlying math that enables the in-game physics world,

such as calculating gravity, movements, or enemy chasing behaviors.

Additionally, you may not consider physical interaction in a mathematical

manner, such as collisions between different objects and the reflections

of these objects after they collide. These underlying mathematical

computations are critical to implementing a successful video game. When

creating a game, whether you intend on using a game engine or you intend

on performing the computations yourself, understanding the details and

knowing how the underlying mathematics work and when to use them to

create what you want, where you want, is vital.

Traditionally math is taught without any application contexts.

Typically, theories are developed based on abstract symbols, formulas are

derived to support these theories, and then numbers are used to verify

the formulas. You are tested on whether you can generate the correct

solution based on how the formulas are applied. It is believed that learning

math in this manner has the benefit of granting the learner the ability

to understand the concepts being taught at the pure abstraction level.

Then, once understood, the application of these concepts to different

disciplinary contexts becomes straightforward. For many learners, this

assumption is certainly true. However, for other types of learners, it can be

difficult to appreciate the intricate details in the abstract without concrete

examples or applications to build off. This fact is recognized by educators

and often story problems are introduced after a basic understanding is

established to help learners gain insights and appreciate the formulas. This

learning approach is taken on and exploited in the context of linear algebra

and video games.

This book takes you on the journey of learning linear algebra, a

branch of mathematics that is the foundation of interactive graphical

applications, like video games. While the underlying theories can be

abstract and complicated, the application of these theories in graphical

object interactions is relatively straightforward. For this reason, this book

approaches linear algebra topics in a concrete manner, based around

CHAPTER 1 INTRODUCTION ANDLEARNING ENVIRONMENT

game-like examples that you can interact with. Through this book, you will

learn a flavor of linear algebra that is directly applicable to video games

and interactive computer graphics as a whole.

Every math concept presented in this book is accompanied with

concrete examples that you can interact with and are relevant to video

game development. It is the intent of this book that you will learn and

know how to apply the concepts in solving the problems you are likely to

encounter during game development. A direct consequence of this focused

approach is that readers may find it challenging to apply the knowledge

gained throughout this book to other disciplines, like machine learning or

computer vision. For example, the dot product, which will be covered in

Chapter 5, can be used to calculate intersection positions, and it can also

be used in machine learning algorithms as a data reduction tool; however,

this book will only focus on the video game applications of the dot product.

If you are looking for general knowledge in linear algebra, you should

consider a more traditional textbook. Such a book is likely to cover concepts

at levels that are suitable for applications for multiple problem spaces. If you

are interested in solving problems specific to interacting graphical objects,

especially for game development, then this is the perfect book for you.

After the introduction to the game engine and terminologies in this

chapter, Chapter 2 reviews the Cartesian Coordinate System and number

intervals leading to the exploration of one of the most widely used tools

in game development—bounding boxes. Chapter 3 continues bounding

volume exploration by examining bounding spheres while also beginning

the investigation of relationships between positions. Chapter 4 introduces

vectors to formalize the relationships between positions in 3D space and

applies vector concepts in controlling and manipulating object motions

under external effects like wind or current flow. Chapter 5 presents the

vector dot products to relate vectors, represents line segments based on

vectors, and demonstrates the application of these concepts in computing

distances between objects and motion paths when approximating

potential collisions. Chapter 6 discusses the vector cross product,

CHAPTER 1 INTRODUCTION ANDLEARNING ENVIRONMENT

examines the space that defines vectors, defines vector plane equation,

and illustrates the application of these concepts in computing intersections

and reflections of moving objects and 2D planes. Finally, Chapter 7

summarizes all of the concepts presented in an aggregated example.

Choice ofUnity Engine

Unity is the choice of platform for presenting the mathematical concepts

covered in this book for three reasons. First, Unity provides elaborate

utilities and efficient support for its user to implement and visualize

solutions based on mathematical formulas. Its Application Programming

Interface (API) implements the basic and many advanced linear algebra

functionalities, while the Entity-Component-System (ECS) game object

architecture allows straightforward user scripting. These qualities give

Unity a close pairing of math concepts to your programming code,

assisting in the visualization of the mathematical solution that you are

trying to understand. This close pairing cannot be understated and is the

backbone of this book.

The second reason for choosing Unity is that, being a game engine, the

system allows for a high degree of intractability with the solution as well as

the ability to visualize that solution. For example, in addition to being able

to examine the results of a ray and 2D plane intersection computation in

real time, you will also be able to manipulate the ray and the 2D plane to

observe the effects on the intersection. The ability to interact, manipulate,

and examine the application of mathematical concepts in real time will

give you a greater understanding and appreciation for that concept.

Finally, Unity is chosen because there is no better way to learn math

concepts for video games than through a popular game engine!

While this book is meant for readers who may be interested in building

a video game in Unity, the focus of this book is on the math concepts and

their implementations and not on how to use Unity. This book teaches the

CHAPTER 1 INTRODUCTION ANDLEARNING ENVIRONMENT

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