《計(jì)算機(jī)圖形學(xué)教學(xué)資料》ppt課件

1、4.2 3D transformations nTranslate(平移) transformationsnScale(縮放) transformationsnShear(錯(cuò)切) transformations nRotate(旋轉(zhuǎn)) transformationsnReflect(反射) transformationsnComposition(復(fù)合) of 3D transformationsn與二維平移變換類(lèi)似地運(yùn)用齊次坐標(biāo)表示為：110001000100011zyxtttzyxzyx記為：PTP其中參 數(shù)zyxttt,是 平 移 距 離Translate transformation11

2、zyxtzztyytxx記為：PSP110000000000001zyxssszyxzyxScale transformationnAbout origin11zszysyxsxzyxCont.nAbout arbitrary point),(fffzyxThe arbitrary reference point is :Cont.nAbout arbitrary pointtranslate, scale about origin, inverse translatenConsists of:),(fffzyxnThe arbitrary reference point is :Cont.那

3、么變換矩陣為:),(),(),(fffzyxfffzyxTsssSzyxTT1000)1 (00)1 (00)1 (00fzzfyyfxxzssyssxssShear transformationsnDependence axis(依賴(lài)軸: corresponding coordinate is remained nDirection axis方向軸: corresponding coordinate is changed linearly nRepresentations:zcyzyyayxxY為依賴(lài)軸：zzbzyyazxxz為依賴(lài)軸：zcxzybxyxxX為依賴(lài)軸：n變換的普通表達(dá)式是：S

4、hear transformations11212121zycxczzbyxbyzayaxx1000010101212121ccbbaaSHnParameters: rotate axis, rotate anglen二維旋轉(zhuǎn)變換是三維空間中繞Z軸的旋轉(zhuǎn)11000010000cossin00sincos1zyxzyx記為：PRPz)(XYZRotate transformationRotate about X axisEqually with changing the coordinate system x,y,z to the coordinate system y,z,x. 110000c

5、ossin00sincos000011zyxzyxPRPX)(YZXXYZRotate about Y axisChanging system x,y,z to system z,x,y110000cos0sin00100sin0cos1zyxzyxZXYXYZ?:about arbitrary linen是關(guān)于某直線或平面進(jìn)展的n關(guān)于某個(gè)軸進(jìn)展的反射變換等同于關(guān)于該軸做180度的旋轉(zhuǎn)變換nFor instance: about Z axis1000010000100001TReflect transformation反射變換?:about arbitrary symmetry axisCon

6、t.n當(dāng)反射平面是坐標(biāo)平面時(shí)，等同于進(jìn)展左、右手坐標(biāo)系的互換，相應(yīng)變換矩陣是把第三維坐標(biāo)值取反nFor instance: about XOY plane1000010000100001T?About arbitrary symmetry planenFor instance: rotating about arbitrary line nOverlapping arbitrary line with Z axisnResolving a series of problems nReflect about an arbitrary symmetry linenReflect about an

7、arbitrary symmetry plane Composition transformationsn旋轉(zhuǎn)軸不與坐標(biāo)軸重合時(shí)變換的實(shí)現(xiàn)：n經(jīng)復(fù)合變換使旋轉(zhuǎn)軸與某坐標(biāo)軸重合n繞指定軸進(jìn)展旋轉(zhuǎn)變換n復(fù)原坐標(biāo)系假設(shè)給定旋轉(zhuǎn)軸),;,( :22211121zyxzyxPP和旋轉(zhuǎn)角 YZXP1P2Rotate about arbitrary line1translate P1 to overlap origin10001000100011111zyxT無(wú)妨設(shè)P1P2為方向單位矢量，P2點(diǎn)坐標(biāo)為(a,b,c)YZxP1P2YZxCont.cbaXYZOP1P2XYZCont.Cont.cbaP1P2X

8、YZ(2)rotate about X axis to put the line on XOZcbaXYZCont.(2)rotate about X axis to put the line on XOZcbaXYZCont.(2)rotate about X axis to put the line on XOZcbaXYZCont.(2)rotate about X axis to put the line on XOZcbaXYZCont.(2)rotate about X axis to put the line on XOZcbaXYZCont.(2)rotate about X

9、axis to put the line on XOZcbaXYZCont.(2)rotate about X axis to put the line on XOZcbaXYZ2222sincoscbbcbcCont.(2)rotate about X axis to put the line on XOZ22100000000001cbdwhereasdcdbdbdcRxThen P2 is (a,0,d)Transformation matrix(變換矩陣cbaXYZ), 0 ,(daCont.(3) Rotate about Y axis to overlap the line wit

10、h Z axisXYZadsincosCont.100000001000daadRyXYZ(4) Rotate about Z axis namely the line through )(2zRTCont.XYZP1P2Cont.(4)recover the coordinate systemThe final transformation is: R()=T1-1Rx-1(-)Ry-1() Rz()Ry()Rx()T1Cont.n關(guān)于恣意直線(或平面)的反射可以分解為平移、旋轉(zhuǎn)使得指定的反射直線或平面與某坐標(biāo)軸或平面重合和關(guān)于坐標(biāo)直線(或坐標(biāo)平面)的反射，再加恢復(fù)變換。Exercises

11、out classroomExercise 4.11Given a unit cube with one corner at (0,0,0) and another opposite corner at (1,1,1),derive the transforations necessary to rotate the cube by degree about the main diagonal(對(duì)角線 (from( 0,0,0) to (1,1,1) in the counterclockwise direction when looking along the diagonal toward

12、 the origin. Exercises out classroomExercise 4.14An object is to be scaled by a factor S in the direction whose direction cosines are (,).Derive the transformation matrix .Two methods of transformationnCoordinate system fixed, Graphics changednGraphics fixed, Coordinate system changed (1)坐標(biāo)系不變,圖形變換;

13、 (2)圖形不變,坐標(biāo)系變換.變換的兩種實(shí)現(xiàn)方法:Transforming coordinate systemnTwo means:nDefine the new coordinate system directlynDefine a vector in y direction of the new coordinate systemCont.1. Define a new system: composition of transformations(x0,y0)(1) translate: T-x0，-y0(2) rotate:R(-)(3) scale(4) composition of

14、above transformations (notice the sequence)Cont.nThe matrix is:),()(00yxTR100)cos()sin()sin()cos(00yxnamely：Cont.2. Define a vector in y direction of new system:Y axis is: (x0,y0)(x1,y1),(0101yxyvvPPPPX axis is:),(),(yxxyxuuvvTransformation is:10000yvvxuuyxyxContrast (x0,y0)100cossinsincos00yx(x0,y0

15、)(x1,y1)10000yvvxuuyxyxVS.XYZXYZ新坐標(biāo)系的原點(diǎn)坐標(biāo)是),(000zyx，三個(gè)坐標(biāo)軸的單位向量分別是),(),(),(zyxzyxzyxnnnvvvuuu nTransform from an old coordinate system to another new coordinate systemnThe new system is shown in the right figure:Mode transformationCont.nComposition of translation and rotation:當(dāng)坐標(biāo)系運(yùn)用不同的縮放時(shí)，還需定義縮放補(bǔ)償。10

16、001000100011000000000zyxnnnvvvuuuTzyxzyxzyx4.3 window-to-viewport transformationnWorld Domain(用戶(hù)域WD)n指程序員用來(lái)定義草圖的整個(gè)自然空間.nWorld-coordinate system(用戶(hù)坐標(biāo)系WC).n世界坐標(biāo)系n右手直角坐標(biāo)系nWindow(窗口區(qū)W)n在用戶(hù)坐標(biāo)系(世界坐標(biāo)系WC中預(yù)先選定的將產(chǎn)生圖形顯示的區(qū)域稱(chēng)為窗口Related conceptsCont.nScreen Domain(屏幕域SD)n設(shè)備輸出圖形的最大區(qū)域,是有限的整數(shù)域.nViewport(視圖區(qū)V)n在顯示器坐標(biāo)

17、系中規(guī)定的顯示圖形的區(qū)域稱(chēng)為視圖區(qū).nScreen coordinates屏幕坐標(biāo)系n(normalized) device coordinatesndevice coordinates: addressing by pixelsnNDC: -1，1-a，a窗口的取景器作用Window as a viewfinder利用窗口尺寸變化改動(dòng)顯示圖形的大小 選窗口的視見(jiàn)變換選窗口的視見(jiàn)變換Cont.n視見(jiàn)變換將用戶(hù)坐標(biāo)系中窗口內(nèi)的圖形變換到顯示器中的視見(jiàn)區(qū)中產(chǎn)生顯示.Window-to-Viewport transformationwindowWxlWxrWybWytP(x,y)VxlVxrview

18、portVybVytP(x,y)VxlVxrVxlxWxlWxrWxlxVybVytVybyWybWytWybyWybWytVybVytWybVybyWybWytVybVytyWxlWxrVxlVxrWxlVxlxWxlWxrVxlVxrxCont. WybSVybdyWybWytVybVytSWxlSVxldxWxlWxrVxlVxrSyyxx,設(shè)dyySydxxSxyx則Cont.1100001yxdysdxsyxyxtransform matrix窗口WxlWxrWybWytWxlWxrWybWyt-VxlVxrVybVyt-VxlVxrvybVytCont.VxlVxrVybVytNDC-to-DC transformationnNDC: -1，1-a，anDC: 0，M-10，N