大三05计算机图形学-4647

1、Transformations变换原著Ed AngelProfessor of Computer Science, Electrical and Computer Engineering, and Media ArtsUniversity of New Mexico编辑 武汉大学计算机学院图形学课程组ObjectivesIntroduce standard transformations标准变换Rotation旋转Translation平移Scaling缩放Shear错切Derive homogeneous coordinate transformation matrices齐次坐标变换矩阵L

2、earn to build arbitrary transformation matrices from simple transformations从简单变换创建任意变换矩阵Outline4.6 Affine transformation仿射变换4.7 Translation平移、Rotation旋转、Scaling放缩4.8 Transformation in homogeneous coordinates齐次坐标坐标表示的变换4.9 Concatenation of Transformation变换的复合4.6 Affine transformation仿射变换(General Tran

3、sformations一般变换)A transformation maps points to other points and/or vectors to other vectors所谓变换就是把点映射到其它点，把向量映射到其它向量Q=T(P)v=T(u)Figure 4.33 Transformation54.6 Affine transformation仿射变换(连续变换)直线的像为一条连续曲线在一般变换下，直线的像是由直线上每个点的像构成的，像一般不再是一条直线直线的像不再是直线当变换是连续的时候，直线的像就是一条连续曲线4.6 Affine transformation仿射变换(线性

4、变换)线性变换条件：f(p+q)=f(p)+f(q)可以表示为矩阵相乘的形式：v=ATu线性变换：标架的变换顶点在同一个标架里的变换仿射变换是具有线性不变性的变换4.6 Affine Transformations仿射变换Line preserving保持共线性Characteristic of many physically important transformations物理上重要变换的特征Rigid body transformations: rotation, translation刚体变换：旋转、平移Scaling, shear放缩、错切Importance in graphics

5、 is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints图形学中的重要性在于，此时我们只需要变换线段的两个端点，而由系统自动画出变换后两个端点间的线段为什么需要变换?变换的作用之一构造三维场景雪花的构造变换的作用之二变换的作用之三4.6 Affine Transformations仿射变换(Pipeline Implementation流水线实现)Transformation变换Rast

6、erizer光栅化 uv u vTT(u)T(v)T(u)T(u)T(v)T(v)Vertices顶点Vertices顶点pixelsFrame buffer帧缓冲区(from application program来自于应用程序)4.6 Affine Transformations仿射变换(Notation记号-1)We will be working with both coordinate-free representations of transformations and representations within a particular frame在后面的处理中，既会考虑变换的

7、坐标无关的表示，也会考虑变换在特定标架下的表示 P,Q, R: points in an affine space仿射空间中的点 u, v, w: vectors in an affine space仿射空间中的向量 a, b, g: scalars标量 4.6 Affine Transformations仿射变换(Notation记号-2)p, q, r: representations of points点的表示 -array of 4 scalars in homogeneous coordinates在齐次坐标中为由四个标量构成的数组 u, v, w: representations

8、of vectors向量的表示 -array of 4 scalars in homogeneous coordinates在齐次坐标中为由四个标量构成的数组4.7 Translation平移、Rotation旋转、Scaling放缩(4.7.1 Translation平移)Move (translate, displace) a point to a new location把一个点移到新的位置Displacement determined by a vector d平移由一个向量d确定Three degrees of freedom三个自由度P=P+dPPd4.7.1 Translatio

9、n平移(How many ways?对象的平移) Although we can move a point to a new location in infinite ways多种方法, when we move many points there is usually only one way通常一种方法objecttranslation: every point displaced by same vector把一个对象上的所有点沿同一向量平移4.7.1 Translation平移(Translation Using Representations平移的表示)Using the homog

10、eneous coordinate representation in some frame应用在某个标架中的齐次坐标表示 p= x y z 1T p=x y z 1T d=dx dy dz 0THence p = p + d or x=x+dx y=y+dy z=z+dznote that this expression is in four dimensions and expresses point = vector + point 注意：这个表达式是四维而且表示的点= 点+ 向量4.7.1 Translation平移(Translation Matrix平移矩阵)We can also

11、 express translation using a 4 x 4 matrix T in homogeneous coordinates可以用在齐次坐标中一个44的矩阵T表示平移： p=Tp where T = T(dx, dy, dz) =This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together这种形式更容易实现，因为所有的仿射变换都

12、可以用这种形式表示，矩阵乘法可以复合在一起4.7.2 Rotation旋转(Rotation (2D)二维旋转)Consider rotation about the origin by q degrees考虑绕原点旋转度radius stays the same, angle increases by q半径保持不变，角度增加了x=x cos q y sin qy = x sin q + y cos qx = r cos fy = r sin fx = r cos (f + q)y = r sin (f + q)4.7.2 Rotation旋转4.7.2 Rotation 三维旋转4.7.2

13、 Rotation (Rotation about the z axis绕 Z 轴的旋转)Rotation about z axis in three dimensions leaves all points with the same z 考虑绕z轴旋转，点的z坐标不变Equivalent to rotation in two dimensions in planes of constant z等价于在 z=常数的平面上进行二维旋转or in homogeneous coordinates其齐次坐标表示为 p = Rz(q) px=x cos q y sin qy = x sin q + y

14、 cos qz =z4.7.2 Rotation (Rotation Matrix旋转矩阵)R = Rz(q) =4.7.2 Rotation(Rotation about x and y axes绕x轴和y轴的旋转)Same argument as for rotation about z axis与绕z轴的旋转完全类似For rotation about x axis, x is unchanged对于绕x轴的旋转，x坐标不变For rotation about y axis, y is unchanged对于绕y轴的旋转，y坐标不变R = Rx(q) =R = Ry(q) = 刚体变换4

15、.7.3 Scaling放缩S = S(sx, sy, sz) =x=sxxy=syyz=szzp=SpExpand or contract along each axis (fixed point of origin)沿每个坐标轴伸展或收缩(原点为不动点)缩放矩阵 S均匀与非均匀缩放4.7.3 Scaling放缩 放缩因子4.7.3 Scaling放缩(Reflection反射)特例: corresponds to negative scale factors对应于负的放缩因子originalsx = -1 sy = 1sx = -1 sy = -1sx = 1 sy = -14.7 Tra

16、nslation平移、Rotation旋转、Scaling放缩(Inverses逆变换)Although we could compute inverse matrices by general formulas, we can use simple geometric observations虽然可以直接计算矩阵的逆，但根据几何意义可以给出各种变换的逆Translation平移: T-1(dx, dy, dz) = T(-dx, -dy, -dz) Rotation旋转: R -1(q) = R(-q)Holds for any rotation matrix对任一旋转矩阵成立Note th

17、at since cos(-q) = cos(q) and sin(-q)=-sin(q)R -1(q) = R T(q)Scaling放缩: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz) 4.8.4 Shear错切Helpful to add one more basic transformation另外一种实用的基本变换Equivalent to pulling faces in opposite directions等价于把面向相反方向倾斜4.8.4 Shear错切(Shear Matrix错切矩阵)Consider simple shear along x

18、 axis考虑沿x轴的错切x = x + y cot qy = yz = zH(q) = 4.9 Concatenation of transformation (Concatenation变换的复合)We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices可以通过把旋转、平移与放缩矩阵相乘从而形成任意的仿射变换Because the same transformation is applied to many

19、vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p 由于对许多顶点应用同样的变换，因此构造矩阵M = ABCD的代价相比于对许多顶点p计算Mp的代价是很小的The difficult part is how to form a desired transformation from the specifications in the application难点在于如何根据应用程序的要求构造出满足要求的变换

20、矩阵例如 围绕空间任意轴的三维旋转 ？4.9 Concatenation of transformation(Order of Transformations变换的顺序)Note that matrix on the right is the first applied注意在右边的矩阵是首先被应用的矩阵Mathematically, the following are equivalent从数学的角度来说，下述表示是等价的 p = ABCp = A(B(Cp)变换的顺序是不可交换的Note many references use column matrices to represent points. In terms of column matrices列矩阵 pT = pTCTBTAT4.9.1 Rotation About a Fixed Point other than the Origin绕不同于原点的固定点旋转Move fixed point to origin把固定点移到原点Rotate旋转Move fixed point back把固定点移回到原来位置 M = T(pf) R(q) T(-p