1、Chapter 4 Geometrical Transformations1.Vector(矢量2.Matrix矩陣Mathematics Preparation for transformation1兩個(gè)矢量之和：矢量運(yùn)算(Vector)2兩個(gè)矢量的點(diǎn)積：21212121zzyyxxVV),(21212121zzyyxxVV3矢量的長度：2111111121111)()(zzyyxxVVV矢量運(yùn)算4兩個(gè)矢量的叉積：),(),(12211221122122112211221122211121yxyxxzxzzyzyyxyxxzxzzyzyzyxzyxkjiVVNoImagemnmmnnaaaa

2、aaaaaA212221111211aij是矩陣中第i行第j上的元素。上述矩陣記為A，或AmXn，或(aij) mXn。矩陣(matrix)Cont.ndefinition：mnmmnnmnaaaaaaaaaA212222111211m=n時(shí)：A稱為方陣m=1時(shí)：退化為行向量(vector)n=1時(shí)：退化為列向量(vector) 時(shí)：稱為A=Bijijba n nm mn na aa aa aA A11211 Cont.n單位矩陣：100010001nnIn零矩陣：0000000000mnICont.n矩陣加法：mnmnmnCBAijijijbac其中：n矩陣加法滿足的運(yùn)算律：n結(jié)合律n交換律

3、nA+0=ACont.n矩陣數(shù)乘mnmmnnmnijmnkakakakakakakakakakakA212222111211)(Cont.n矩陣乘法：mpnpmnCBAnlljilijbac1其中：n矩陣乘法滿足的運(yùn)算律：n結(jié)合律n分配律nAI=IA=An不滿足交換律！Cont.n矩陣轉(zhuǎn)置nmmnnnmmTmnBaaaaaaaaaA212221212111n矩陣轉(zhuǎn)置運(yùn)算律：TTTTTTTTTTABBAkAkABABAAA)()4()()3()()2()() 1 (Cont.n矩陣行列式n非奇特矩陣n逆矩陣：非奇特矩陣具有逆矩陣IAAAA11假設(shè)把n維空間坐標(biāo)叫做普通坐標(biāo),那么相應(yīng)的n+1維空間

4、坐標(biāo)叫做齊次坐標(biāo). 齊次坐標(biāo)技術(shù):用n+1維向量表示n維向量的技術(shù)ordinary coordinatehomogeneous coordinate(x，y)(hx，hy，h)其中，h是一個(gè)恣意的非零標(biāo)量齊次坐標(biāo)(homogeneous coordinate)齊次坐標(biāo)(homogeneous coordinate)0),(),(0),(),(hhhzhyhxzyxhhhyhxyx1husually那么 (x，y)的齊次坐標(biāo)為(x，y，1)Cont.xyW The XYW homogeneous coordinate space, with the W = 1 plane and point P

5、( X, Y, W ) projected onto the W = 1 plane.PRepresentation of 2D transformations1001232221131211yxcycxcycycxcx11001232221131211yxccccccyxRotate, scale, symmetry, sheartranslateEntire scaleproject代數(shù)式矩陣式3D spacen右手直角坐標(biāo)系XYZnPoints: zyxPAlgebraic representation of 3D transformations100013433323124232221

6、14131211zyxazayaxazazayaxayazayaxax),( :),( :zyxafterzyxbefore；Matrix representation1144434241343332312423222114131211zyxaaaaaaaaaaaaaaaazyxnIntroduce homogeneous coordinate0 0 0 1變換的兩種實(shí)現(xiàn)方法: (1)坐標(biāo)系不變,圖形變換; (2)圖形不變,坐標(biāo)系變換.Transformation method4.1 2D transformations nTranslate(平移) transformationsnRotat

7、e(旋轉(zhuǎn)) transformationsnScale(縮放) transformationsnReflect(反射) transformationsnShear(錯(cuò)切) transformationsnComposition(復(fù)合) of 2D transformations 在二維坐標(biāo)系中,將點(diǎn)P(x,y)在x、y軸方向分別平移tx、ty，得到點(diǎn)P(x,y)，那么P點(diǎn)與P點(diǎn)的坐標(biāo)關(guān)系為： yxtyytxx變換矩陣:T=yxtt矢量方式為：P=P+TTranslate transformations(平移變換)Before translationAfter translation(4,5)(

8、7,5)xy(7,1)(10,1)xy yxttyxyxCont.用齊次坐標(biāo)表示平移變換過程：.1001001,1,1 yxttTyxPyxPPTP Whereas:Translate matrixCont.nTranslating an object: translate is an rigid-body transformationnTranslating every point of the objectnTranslating the key-point of the object and re-defining the objectnConverse transformation:1

9、0010011yxttTP(x,y)P(x,y)OrP1P1)sin()cos(ryrxsincosryrxRotate transformations(refer to origin)sinsincoscosrrsincoscossinrrsincossincosxyyyxxsincossincosxyyyxxCont.變換矩陣：1000cossin0sincos)(RPRP)( 那么變換公式為：vPositive : anti-clockwise rotate(逆時(shí)針) vNegative : clockwise rotate(順時(shí)針)逆變換逆變換:)(RCont.nRotating an

10、 object圖元的旋轉(zhuǎn)變換n旋轉(zhuǎn)變換是剛體變換nRotating every point of object through the same anglenBe rigid-body transformations: rotating the key-point and re-defining the objectCont.Rotation of a house. The house also changes positionBefore rotation(9,2)(5,2)xyAfter rotationxy(2.1,4.9)(4.9,7.8)固定某個(gè)點(diǎn)的旋轉(zhuǎn)變換?Scaling tran

11、sformations1ysyxsxyx變換矩陣:1000000yxssS變換公式:1),(1yxssSyxyx縮放變換是指對點(diǎn)的縮放變換是指對點(diǎn)的X，Y坐標(biāo)值進(jìn)展縮放。坐標(biāo)值進(jìn)展縮放。Sx , Sy 稱為縮放系數(shù)，可取任何正數(shù); S稱為縮放矩陣?？s放變換可使物體產(chǎn)生重定位，如右圖所示，縮放比例不同，定位間隔也不同。當(dāng)縮放系數(shù)大于1時(shí)，物體被放大，否那么減少；當(dāng)SxSy時(shí)，物體發(fā)生等比變換，否那么發(fā)生差值縮放，產(chǎn)生變形。Scaling transformations2P0P1OP1P0Scaling an objectnPolygon多邊形nScaling the vertexes and t

12、hen re-defining the polygonncircle圓中心對稱圖形nScaling the radiusnPrimitives defined by some parameters給定定義參數(shù)的圖形nScaling the parameters and re-defining the primitivesyyxx相對X軸對稱：Symmetry transformations變換矩陣:100010001xSY變換公式:11yxSYyxx對稱變換是產(chǎn)生物體鏡象的一種變換.Cont.相對Y軸對稱:11000100011yxyxyyxxCont.11000100011yxyx相對原點(diǎn)對

13、稱:yyxxCont.11000010101yxyx11000010101yxyx相對y=x直線對稱:相對y=-x直線對稱:xyyxxyyx錯(cuò)切變換：堅(jiān)持圖形上各點(diǎn)的某一坐標(biāo)值不變，錯(cuò)切變換：堅(jiān)持圖形上各點(diǎn)的某一坐標(biāo)值不變，而另一坐標(biāo)值關(guān)于該坐標(biāo)值呈線性變化。而另一坐標(biāo)值關(guān)于該坐標(biāo)值呈線性變化。坐標(biāo)堅(jiān)持不變的坐標(biāo)軸稱為倚賴軸，其它坐標(biāo)坐標(biāo)堅(jiān)持不變的坐標(biāo)軸稱為倚賴軸，其它坐標(biāo)軸稱為方向軸。軸稱為方向軸。(1)以以y軸為依賴軸的錯(cuò)切變換軸為依賴軸的錯(cuò)切變換(沿沿x方向的錯(cuò)切方向的錯(cuò)切 即即y不變，不變，x的值隨的值隨y的值而線性變化的值而線性變化yyyshxxxShx=tan Shear tran

14、sformationsyxshyxxyShy=tan (2)以x軸為依賴軸的錯(cuò)切變換Cont.Cont.nTwo kinds:nShear along x axisnShear along y axisnGeneral representation of shear:11bxydyxyxb=0 or d=011000101yxbd變換合成n根據(jù)矩陣運(yùn)算的性質(zhì)，可推出二維根本變換的如下性質(zhì)：n平移變換和旋轉(zhuǎn)變換具有可加性n縮放變換具有可乘性),(),(),(21211122yyxxyxyxttttTttTttT)()()(2112RRR),(),(),(21211122yyxxyxyxssssS

15、ssSssSRotate transformation about arbitrary point),()(),(0000yxyxTRTT總),(yx),(yx),(yx),(yx),(00yx),(yx),(yx),(00yxPTPyx),(00)( PRP ),( 00PTPyx),(00yxSuppose the point is then the transformation can be composed by some fundamental transformations Translation(T)Rotation(R)Inverse translationT=T-1R TCo

16、nt.Scaling about arbitrary pointnReference point: , fix up point before and after scaling),(refrefyxComposition of translate, scale about origin, and inverse translate transformationsCont.1100)1 (0)1 (01yxsyssxsyxyrefyxrefxNamely:PTSTPrefrefyxrefrefyxssyx),(),(),(Symmetry about arbitrary lineTRSYRTTX11總TRSY1R1T作業(yè)：關(guān)于恣意直線的對稱變換直線為 y= x+b寫出 T總33Exercises nExercise 4.1 nProve that we can transform a line by transforming its endpoints and then constructing a new line between the transformed endpoints.n經(jīng)過變換后的兩個(gè)端點(diǎn)可定義變換后的直線Exercises nExe