Zed3D：实时3D图形编程精要

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"Zed3D - A Compact Reference for 3D Computer Graphics Programming by Sébastien Loisel" 本文档是Sébastien Loisel编写的关于3D计算机图形编程的精简参考指南，最初源于作者的工作笔记，经过修订和完善，成为了一个适用于初学者和进阶者的实用参考资料。Zed3D主要关注实时3D图形，涵盖了3D图形学的多个方面，旨在以清晰、实用和全面的方式解释相关概念和技术。在文档中，读者将能够了解到： 1. **理论基础**：3D图形学的基础理论，包括坐标系统、向量和矩阵运算、几何变换（如平移、旋转和缩放）、投影（透视投影和平行投影）以及光照模型等。 2. **图形管线**：3D图形渲染的核心流程，从顶点数据到屏幕像素的转换，涉及顶点处理、裁剪、视口变换、光栅化和颜色计算等步骤。 3. **数学工具**：用于3D图形编程的重要数学工具，如线性代数（向量和矩阵）、四元数用于表示旋转，以及三角函数在几何计算中的应用。 4. **纹理映射**：如何将2D纹理应用到3D模型表面，增加细节和真实感，包括纹理坐标、纹理过滤和纹理坐标的生成方法。 5. **渲染技术**：包括深度缓冲（Z-buffer）用于解决遮挡问题，以及各种着色技术，如Gouraud着色和Phong着色。 6. **图形API**：虽然没有具体提及，但可能会讨论到像OpenGL或Direct3D这样的图形库，它们是实现3D图形编程的常用接口。 7. **性能优化**：针对实时3D图形，如何通过优化算法和数据结构来提高渲染速度，例如顶点缓存、硬件加速和多线程渲染。 8. **实例和代码**：文档可能包含示例代码和实践案例，帮助读者理解和应用所学知识，例如基本的3D物体绘制、场景管理或交互式应用开发。尽管作者自谦为学生，但这份资料已展现出对3D图形学的深入理解和实践经验，适合希望学习或巩固3D图形编程技能的读者。随着版本的迭代，这份工作笔记逐步发展成为一个系统化的教程，对于初学者和专业人士来说都是有价值的参考资料。

If one wants a plane that does not pass through the origin, one should proceed as

follows. (This uses an intuitive form of affine transformations, described in depth

later). First, find out the displacement vector K that describes the position of the

plane in relation to the origin. Thus, if you subtract K from all the points in the

plane, the plane ends up at the origin, and we can use the definition above. Thus,

the new definition of the plane is A

•

(X-K)=0.

To make this a bit more explicit, let A=(A,B,C) and X=(x,y,z) and K=(k1,k2,k3).

Then the plane equation can be rewritten as: A

(x-k1)+B

(y-k2)+C

(z-k3)=0. A

little algebra allows us to rewrite it as A

x+B

y+C

z=-A

k1-B

k2-C

k3. By

setting D=-A

k1-B

k2-C

k3, we can make one more rewrite, which is the final

form: A

××

x+B

××

y+C

××

z=D.

It is important to remember that multiplying both side of the equation by a

constant does not change the plane. Thus, plane x+y+z=1 is the same as plane

2x+2y+2z=2.

Note that in this last representation, (A,B,C) is the normal vector to the plane.

The last equation can also be re-written A

••

X=D. It would also be easy to

demonstrate the following, but I will not do it. For any point P, (A

••

P-D)/|A| is the

signed distance to the plane A

•

X=D. The sign can help you determine on what

side of the plane that the point P lies on. If it is 0, P is on the plane. If it is positive,

P is in the direction that the normal points to. If it is negative, P is on the side

opposite of the normal. This has application in visible surface determination

(namely, back face culling).

Also note that if |A|=1, then the signed distance equation simplifies to A

•

P-D.

It is easy to demonstrate that the equation for a line in n-space, for any integer

value of n>0, is t

××

U+W, where U is a vector parallel to the line and W is a point

on the line. As t takes all real values, we generate a line.

Exercises

Q1 - Given the basis U=(1,3,2) and V=(2,2,2), and the position vector W=(1,1,0),

find the position in 3d space of the point (3,2) in UV space.

Q2 - Express the plane described in Q1 in the form Ax+By+Cz=D

Q3 - Find the signed distance of point (4,2,4) to the plane using the answer for

question 2.

Q4 - Given two basis vectors for a plane, P and Q, in 3d space, and a position

vector for the plane, R, plus the direction vector of a line, M, that passes through

origin, find the pq space point of intersection between the line and the plane.

Matrix addition is defined as follows. Given 2 matrices A=(a

) and B=(b

) of

same dimension pxq, then U=(u

)=A+B is defined as being (u

)=(a

Matrix multiplication by a scalar is defined also as follows. Given the matrix M

and a scalar k, then the operation U=(u

)=k

M is defined as u

Matrix multiplication is a bit more involved. It is defined using sums, as follows.

Given matrix A of dimension pxq, and matrix B of dimension qxr, the product

C=AxB is given by:

∑

≤

)

More explicitly, for example, we have, for A and B 2x2 matrices:

(Note:

∑

≤

) means "sum of (a

) for k varying from 1 to q.")

It is important to notice that matrix multiplication is not commutative in the

general case. For example, it is not true that A

B=B

A with A and B matrices in

the general case, even if A and B are square matrices. Matrix multiplication is,

however, associative (ie, A

C)=(A

C) and distributive (ie,

A(B+C)=AB+AC).

The identity matrix has the property that, for any matrix A, A

I=I

A=A (I is the

neutral element of matrix multiplication).

Matrix transposition of matrix A, noted A

, reflects the A matrix along the great

diagonal. That is, say A=(a

) and A

=(b

), then we have b

There are also other interesting operations you can do on a matrix, however they

are much, much more involved. As of now, I am not willing to get too deeply into

this. The topics of interest are matrix determinant (which has a recursive

definition) and matrix inversion. I will content myself by giving one definition of

matrix determinant and one way of finding matrix inverse. Note that there are at

least a couple of different definitions for determinant, though they usually boil

down to the same thing. Also, there are many ways of finding the inverse of a

matrix, I will contend myself with presenting only one method. Strict definitions

will be given, for more extensive coverage, consult a linear algebra book.

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