3D游戏与计算机图形学数学原理-第三版

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"Lengyel E.-Mathematics for 3D game programming and computer graphics-PTR (2011)" 是一本专注于3D游戏编程和计算机图形学中数学应用的专业书籍，由Eric Lengyel撰写，是第三版。本书旨在为游戏开发人员和计算机图形学爱好者提供必要的数学基础，帮助他们理解和实现3D图形技术。书中涵盖了一系列关键的数学概念，这些概念在创建复杂3D环境、物体变换、光照计算以及物理模拟等方面起着核心作用。具体知识点包括但不限于： 1. **线性代数**：基础的向量和矩阵运算，如向量加减、标量乘法、矩阵乘法，以及矩阵的逆、转置和行列式。此外，还包括向量的点积和叉积，它们在几何变换中至关重要。 2. **坐标系统和变换**：讲解了局部坐标系和世界坐标系的概念，以及如何通过旋转、平移和缩放矩阵实现对象在3D空间中的位置和姿态改变。 3. **四元数**：四元数用于高效地表示和操作3D旋转，避免了传统欧拉角导致的万向节死锁问题。 4. **几何建模**：包括基本几何形状（如立方体、球体、圆柱体）的构建，以及更复杂的多边形网格和细分表面。 5. **图形渲染**：探讨视锥体裁剪、透视投影和平行投影等基本渲染技术，以及光照模型，如Phong光照模型。 6. **运动学和动力学**：介绍了牛顿运动定律和基本的物理模拟，如刚体动力学和碰撞检测。 7. **曲线和曲面**：NURBS（非均匀有理B样条）和贝塞尔曲线曲面在动画和建模中的应用。 8. **图形API**：虽然不是数学本身，但可能涉及如OpenGL或DirectX等图形库的基本用法，这些API通常需要理解背后的数学原理。 9. **算法和数据结构**：如快速傅里叶变换(FFT)在图像处理中的应用，以及用于优化3D场景的树形数据结构（如KD树）。 10. **程序设计技巧**：如何有效地利用这些数学知识编写高效的代码，例如向量运算的SIMD（单指令多数据）优化。此书适合有一定编程基础的读者，通过深入浅出的讲解和实际案例，帮助读者将理论知识应用于实际游戏开发和计算机图形项目中。它不仅提供了扎实的数学基础，还强调了这些理论在实际工程实践中的重要性和应用方式。

ContentsOverview xv

Chapter 5: Geometry for 3D Engines. It is at this point that we begin to see

material presented in the previous three chapters applied to practical applications

in 3D game programming and computer graphics. After analyzing lines and

planes in 3D space, we introduce the view frustum and its relationship to the vir-

tual camera. This chapter includes topics such as field of view, perspective-

correct interpolation, and projection matrices.

Chapter 6: Ray Tracing. Ray tracing methods are useful in many areas of game

programming, including light map generation, line-of-sight determination, and

collision detection. This chapter begins with analytical and numerical root-

finding techniques, and then presents methods for intersecting rays with common

geometrical objects. Finally, calculation of reflection and refraction vectors is

discussed.

Chapter 7: Lighting and Shading. Chapter 7 discusses a wide range of topics

related to illumination and shading methods. We begin with an enumeration of

the different types of light sources and then proceed to simple reflection models.

Later, we inspect methods for adding detail to rendered surfaces using texture

maps, gloss maps, and bump maps. The chapter closes with a detailed explana-

tion of the Cook-Torrance physical illumination model.

Chapter 8: Visibility Determination. The performance of a 3D engine is heavi-

ly dependent on its ability to determine what parts of a scene are visible. This

chapter presents methods for constructing various types of bounding volumes and

subsequently testing their visibility against the view frustum. Large-scale visibil-

ity determination enabled through spatial partitioning and the use of portal sys-

tems is also examined.

Chapter 9: Polygonal Techniques. Chapter 9 presents several techniques in-

volving the manipulation of polygonal models. The first topic covered is decal

application to arbitrary surfaces and includes a related method for performing

vertex depth offset. Other topics include billboarding techniques used for various

special effects, a polygon reduction technique, T-junction elimination, and poly-

gon triangulation.

Chapter 10: Shadows. This chapter discusses shadow-casting sets and the two

prominent methods for generating shadows in a real-time application, shadow

mapping and stencil shadows. The presentation of the stencil shadow algorithm is

particularly detailed because it draws on several smaller geometric techniques.

xvi Preface

Chapter 11: Curves and Surfaces. In this chapter, we examine of a broad varie-

ty of cubic curves, including Bézier curves and B-splines. We also discuss how

concepts pertaining to two-dimensional curves are extended to three-dimensional

surfaces.

Chapter 12: Collision Detection. Collision detection is necessary for interaction

between different objects in a game universe. This chapter presents general

methods for determining whether moving objects collide with the static environ-

ment and whether they collide with each other.

Chapter 13: Linear Physics. At this point in the book, we begin a two-chapter

survey of various topics in classical physics that pertain to the motion that objects

are likely to exhibit in a 3D game. Chapter 13 begins with a discussion of posi-

tion functions as solutions to second-order differential equations. We then inves-

tigate projectile motion both through empty space and through a resistive medi-

um, and close with a look at frictional forces.

Chapter 14: Rotational Physics. Chapter 14 continues the treatment of physics

with a rather advanced exposition on rotation. We first study the forces experi-

enced by an object in a rotating environment. Next, we examine rigid body mo-

tion and derive the relationship between angular velocity and angular momentum

through the inertia tensor. Also included is a discussion of the oscillatory motion

exhibited by springs and pendulums.

Chapter 15: Fluid and Cloth Simulation. We continue with the theme of phys-

ical simulation by presenting a physical model for fluid motion based on the two-

dimensional wave equation and cloth motion based on a spring-damper system.

Chapter 16: Numerical Methods. The book finishes with an examination of

numerical techniques for calculating trigonometric functions and solving three

particular types of problems. We first discuss effective methods for finding the

solutions to linear systems of any size. Next, we present an iterative technique for

determining the eigenvalues and eigenvectors of a



symmetric matrix. Final-

ly, we study methods for approximating the solutions to ordinary differential

equations.

Appendix A: Complex Numbers.

Although not used extensively, complex

numbers do appear in a few places in the text. Appendix A reviews the concept

of complex numbers and discusses the properties that are used elsewhere in the

book.

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