总结Wikipedia文档
https://www.cnblogs.com/ghj1976/p/5199086.html
0. 知识结构体系
仿射变换
1. 定义
2. 组成
1.定义
core：function that preserves points,straight lines and planes.
特性：
(1)preserves parallel after an affine transformation.
(2)not preserves distances between points and angles between lines, but preserves the ratios of distances between points lying on a straight lines.
2.组成
常见的仿射变换有
translation 平移
scaling 尺度变换
homothety
similarity transformation
reflection
rotation
shear mapping
compositions of all above
wikipedia中针对上述仿射变化的介绍从形象的介绍（带图片），表达式，矩阵表示，齐次线性坐标系下的矩阵表示
2.1 translation
定义：
核心词汇：same distance.
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.
数学表示：
A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.
矩阵表示：
欧式空间中无法表示这种向量相加的变换（A translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point. ），因此采用齐次线性坐标系中的矩阵来表示。
Write the 3-dimensional vector w = (wx, wy, wz) using 4 homogeneous coordinates as w = (wx, wy, wz, 1)
2. scaling 尺度变换
2.1 定义
uniform scaling(isotropic scaling)
non-uniform scaling(anisotropic scaling) 变形，伸缩
尺度变换的方向和尺度变换的大小
2. 矩阵表示
unifom scalingIing 对角阵，变换尺度为v，则矩阵为vI。I为单位阵。
non-uniform scaling 对称矩阵，eigenvectors and eigenvalues！特征值为变换尺度，特征向量为尺度变换的方向。
作为属于不同本征空间的两个或更多个非零向量的组合的向量将朝向具有最大特征值的本征空间倾斜。
3. 齐次线性坐标系矩阵表示
3. homothety 位似变换
4. Similarity 相似变换
定义：
更确切地说，可以通过均匀缩放（放大或缩小），可能通过额外的平移，旋转和反射从另一个获得。
Two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. A modern and novel perspective of similarity is to consider geometrical objects similar if one appears congruent to the other when zoomed in or out at some level.
5. reflection 镜像变换
定义：hyperplane
In mathematics, a reflection (also spelled reflexion)[1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.
6. rotation 旋转变换
Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n− 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.
Mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group (of a particular space). But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the same coordinates. For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations.
正交矩阵！
7. shear mapping 剪切映射
在平面几何中，剪切映射是一个线性映射，它将固定方向上的每个点移位一个与其与该方向平行并穿过原点的线的符号距离成比例的量。[1] 这种类型的映射也称为剪切变换，横向变换或仅剪切。
网友评论