Mathematics for 3D Game Programming and Computer Graphics基础学习

主要结合opengl学习3d的中的数学知识，以后主要精力都集中在底层。

第三章　矩阵

3.1 Matrix Properices

转置矩阵

The transpose of an n × m matrix M, which we denote by M T , is an m × n matrix for which the ( i , j ) entry is equal to M ji (i.e., M ij T = M ji ).

3.2 Linear Systems

线性方程的计算，转换的最终样式。

Finding solutions to a system of linear equations can be achieved by performing elementary row operations on the matrix formed by concatenating the coefficient mattix and the constant vector.

3.3 Matrix Inverses

倒置矩阵的计算

An n * n matrix M is invertile if there exists a matrix, which we denote by M-1, such that M M-1 = I. The matrix m-1 is called the inverse of M.

3.4 Determinants

The determinant of a square matrix is a scalar quantity derived from the entries of the matrix.

3.5 Eigenvalues and Eigenvectors

MV i = λ i V i .

3.6 Diagonalization

矩阵对角化：对n阶矩阵A，如果可以找到可逆矩阵P，使得P-1AP = 对角阵，就称为把方阵A对角化。

第二章　Vectors

2.1 Vector Properties

标量为１的向量和垂直向量。

It should be noted that the term to normalize is in no way related to the term normal vector, which refer to a vector that is perpendicular to a surface at a particular point.

Any vector lying on the same side of plane as P yields a positive dot product with P.

2.2 The Dot Product

求一个向量在另一个向量的投射矩阵计算公式，一横一竖两个向量相乘，将单独的向量Q提取出来。

两个向量点乘等于两个向量的余弦

Vectors whose dot product yields zero are called orthogonal.

2.3 The Cross Product

The cross product of two three-dimensional vectors, also known as the vector product, returns a new vector that is perpendicular to both of the vectors being multiplied together.

2.4 Vector Spaces

N个集合的向量空间可以理解为减去在每一维向量空间的投射。

习题

1. 简单的计算

不同的投射，会产生不同的结果。

-2      [2   1   -6]        [-4/9,    -4/9,   -2/9]

第一章　渲染流水线 (The Rendering Pipeline)

1.1 图像处理器

几何体由顶点和基本几何形状组成，基本几何形状有lines, Triangles, Quads, Polygon等。

The geometrical forms of these objects are each represented by a set of vertices and a particular type of graphics primitive that indicates how the vertices are connected to produce a shape.

Most of the time, the surface of a 3D model is represented by a list of triangles, each of which references three points in a list of vertices.

CPU和GPU工作流程

synchronous operation

The CPU sends rendering commands to the GPU, which then performs the rendering operations while the CPU continues with other task.

交互过程图

1.2　顶点变换 (Vertex Transformation)

坐标系转换

Once a model's vertices have been transformed into camera space, they undergo a projection transformation that has the effect of applying perspctive so that geometry becomes smaller as the distance from the camera increases.

1.3 Rasterization and Fragment Operations