大三05計(jì)算機(jī)圖形學(xué)-4647

1、Transformations變換原著Ed AngelProfessor of Computer Science, Electrical and Computer Engineering, and Media ArtsUniversity of New Mexico編輯 武漢大學(xué)計(jì)算機(jī)學(xué)院圖形學(xué)課程組ObjectivesIntroduce standard transformations標(biāo)準(zhǔn)變換Rotation旋轉(zhuǎn)Translation平移Scaling縮放Shear錯(cuò)切Derive homogeneous coordinate transformation matrices齊次坐標(biāo)變換矩陣L

2、earn to build arbitrary transformation matrices from simple transformations從簡(jiǎn)單變換創(chuàng)建任意變換矩陣Outline4.6 Affine transformation仿射變換4.7 Translation平移、Rotation旋轉(zhuǎn)、Scaling放縮4.8 Transformation in homogeneous coordinates齊次坐標(biāo)坐標(biāo)表示的變換4.9 Concatenation of Transformation變換的復(fù)合4.6 Affine transformation仿射變換(General Tran

3、sformations一般變換)A transformation maps points to other points and/or vectors to other vectors所謂變換就是把點(diǎn)映射到其它點(diǎn)，把向量映射到其它向量Q=T(P)v=T(u)Figure 4.33 Transformation54.6 Affine transformation仿射變換(連續(xù)變換)直線的像為一條連續(xù)曲線在一般變換下，直線的像是由直線上每個(gè)點(diǎn)的像構(gòu)成的，像一般不再是一條直線直線的像不再是直線當(dāng)變換是連續(xù)的時(shí)候，直線的像就是一條連續(xù)曲線4.6 Affine transformation仿射變換(線性

4、變換)線性變換條件：f(p+q)=f(p)+f(q)可以表示為矩陣相乘的形式：v=ATu線性變換：標(biāo)架的變換頂點(diǎn)在同一個(gè)標(biāo)架里的變換仿射變換是具有線性不變性的變換4.6 Affine Transformations仿射變換Line preserving保持共線性Characteristic of many physically important transformations物理上重要變換的特征Rigid body transformations: rotation, translation剛體變換：旋轉(zhuǎn)、平移Scaling, shear放縮、錯(cuò)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圖形學(xué)中的重要性在于，此時(shí)我們只需要變換線段的兩個(gè)端點(diǎn)，而由系統(tǒng)自動(dòng)畫出變換后兩個(gè)端點(diǎn)間的線段為什么需要變換?變換的作用之一構(gòu)造三維場(chǎng)景雪花的構(gòu)造變換的作用之二變換的作用之三4.6 Affine Transformations仿射變換(Pipeline Implementation流水線實(shí)現(xiàn))Transformation變換Rast

6、erizer光柵化 uv u vTT(u)T(v)T(u)T(u)T(v)T(v)Vertices頂點(diǎn)Vertices頂點(diǎn)pixelsFrame buffer幀緩沖區(qū)(from application program來自于應(yīng)用程序)4.6 Affine Transformations仿射變換(Notation記號(hào)-1)We will be working with both coordinate-free representations of transformations and representations within a particular frame在后面的處理中，既會(huì)考慮變換的

7、坐標(biāo)無(wú)關(guān)的表示，也會(huì)考慮變換在特定標(biāo)架下的表示 P,Q, R: points in an affine space仿射空間中的點(diǎn) u, v, w: vectors in an affine space仿射空間中的向量 a, b, g: scalars標(biāo)量 4.6 Affine Transformations仿射變換(Notation記號(hào)-2)p, q, r: representations of points點(diǎn)的表示 -array of 4 scalars in homogeneous coordinates在齊次坐標(biāo)中為由四個(gè)標(biāo)量構(gòu)成的數(shù)組 u, v, w: representations

8、of vectors向量的表示 -array of 4 scalars in homogeneous coordinates在齊次坐標(biāo)中為由四個(gè)標(biāo)量構(gòu)成的數(shù)組4.7 Translation平移、Rotation旋轉(zhuǎn)、Scaling放縮(4.7.1 Translation平移)Move (translate, displace) a point to a new location把一個(gè)點(diǎn)移到新的位置Displacement determined by a vector d平移由一個(gè)向量d確定Three degrees of freedom三個(gè)自由度P=P+dPPd4.7.1 Translatio

9、n平移(How many ways?對(duì)象的平移) Although we can move a point to a new location in infinite ways多種方法, when we move many points there is usually only one way通常一種方法objecttranslation: every point displaced by same vector把一個(gè)對(duì)象上的所有點(diǎn)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應(yīng)用在某個(gè)標(biāo)架中的齊次坐標(biāo)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：這個(gè)表達(dá)式是四維而且表示的點(diǎn)= 點(diǎn)+ 向量4.7.1 Translation平移(Translation Matrix平移矩陣)We can also

11、 express translation using a 4 x 4 matrix T in homogeneous coordinates可以用在齊次坐標(biāo)中一個(gè)44的矩陣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這種形式更容易實(shí)現(xiàn)，因?yàn)樗械姆律渥儞Q都

12、可以用這種形式表示，矩陣乘法可以復(fù)合在一起4.7.2 Rotation旋轉(zhuǎn)(Rotation (2D)二維旋轉(zhuǎn))Consider rotation about the origin by q degrees考慮繞原點(diǎn)旋轉(zhuǎn)度radius stays the same, angle increases by q半徑保持不變，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旋轉(zhuǎn)4.7.2 Rotation 三維旋轉(zhuǎn)4.7.2

13、 Rotation (Rotation about the z axis繞 Z 軸的旋轉(zhuǎn))Rotation about z axis in three dimensions leaves all points with the same z 考慮繞z軸旋轉(zhuǎn)，點(diǎn)的z坐標(biāo)不變Equivalent to rotation in two dimensions in planes of constant z等價(jià)于在 z=常數(shù)的平面上進(jìn)行二維旋轉(zhuǎn)or in homogeneous coordinates其齊次坐標(biāo)表示為 p = Rz(q) px=x cos q y sin qy = x sin q + y

14、 cos qz =z4.7.2 Rotation (Rotation Matrix旋轉(zhuǎn)矩陣)R = Rz(q) =4.7.2 Rotation(Rotation about x and y axes繞x軸和y軸的旋轉(zhuǎn))Same argument as for rotation about z axis與繞z軸的旋轉(zhuǎn)完全類似For rotation about x axis, x is unchanged對(duì)于繞x軸的旋轉(zhuǎn)，x坐標(biāo)不變For rotation about y axis, y is unchanged對(duì)于繞y軸的旋轉(zhuǎn)，y坐標(biāo)不變R = Rx(q) =R = Ry(q) = 剛體變換4

15、.7.3 Scaling放縮S = S(sx, sy, sz) =x=sxxy=syyz=szzp=SpExpand or contract along each axis (fixed point of origin)沿每個(gè)坐標(biāo)軸伸展或收縮(原點(diǎn)為不動(dòng)點(diǎn))縮放矩陣 S均勻與非均勻縮放4.7.3 Scaling放縮 放縮因子4.7.3 Scaling放縮(Reflection反射)特例: corresponds to negative scale factors對(duì)應(yīng)于負(fù)的放縮因子originalsx = -1 sy = 1sx = -1 sy = -1sx = 1 sy = -14.7 Tra

16、nslation平移、Rotation旋轉(zhuǎn)、Scaling放縮(Inverses逆變換)Although we could compute inverse matrices by general formulas, we can use simple geometric observations雖然可以直接計(jì)算矩陣的逆，但根據(jù)幾何意義可以給出各種變換的逆Translation平移: T-1(dx, dy, dz) = T(-dx, -dy, -dz) Rotation旋轉(zhuǎn): R -1(q) = R(-q)Holds for any rotation matrix對(duì)任一旋轉(zhuǎn)矩陣成立Note th

17、at since cos(-q) = cos(q) and sin(-q)=-sin(q)R -1(q) = R T(q)Scaling放縮: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz) 4.8.4 Shear錯(cuò)切Helpful to add one more basic transformation另外一種實(shí)用的基本變換Equivalent to pulling faces in opposite directions等價(jià)于把面向相反方向傾斜4.8.4 Shear錯(cuò)切(Shear Matrix錯(cuò)切矩陣)Consider simple shear along x

18、 axis考慮沿x軸的錯(cuò)切x = x + y cot qy = yz = zH(q) = 4.9 Concatenation of transformation (Concatenation變換的復(fù)合)We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices可以通過把旋轉(zhuǎn)、平移與放縮矩陣相乘從而形成任意的仿射變換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 由于對(duì)許多頂點(diǎn)應(yīng)用同樣的變換，因此構(gòu)造矩陣M = ABCD的代價(jià)相比于對(duì)許多頂點(diǎn)p計(jì)算Mp的代價(jià)是很小的The difficult part is how to form a desired transformation from the specifications in the application難點(diǎn)在于如何根據(jù)應(yīng)用程序的要求構(gòu)造出滿足要求的變換

20、矩陣?yán)?圍繞空間任意軸的三維旋轉(zhuǎn) ？4.9 Concatenation of transformation(Order of Transformations變換的順序)Note that matrix on the right is the first applied注意在右邊的矩陣是首先被應(yīng)用的矩陣Mathematically, the following are equivalent從數(shù)學(xué)的角度來說，下述表示是等價(jià)的 p = ABCp = A(B(Cp)變換的順序是不可交換的Note many references use column matrices to represent points. In terms of column matrices列矩陣 pT = pTCTBTAT4.9.1 Rotation About a Fixed Point other than the Origin繞不同于原點(diǎn)的固定點(diǎn)旋轉(zhuǎn)Move fixed point to origin把固定點(diǎn)移到原點(diǎn)Rotate旋轉(zhuǎn)Move fixed point back把固定點(diǎn)移回到原來位置 M = T(pf) R(q) T(-p