1、4.2 3D transformations,Translate(平移) transformations Scale(縮放) transformations Shear(錯切) transformations Rotate(旋轉(zhuǎn)) transformations Reflect(反射) transformations Composition(復(fù)合) of 3D transformations,與二維平移變換類似地使用齊次坐標(biāo)表示為：,記為：,其中,Translate transformation,記為：,Scale transformation,About origin,Cont.,About

2、 arbitrary point,The arbitrary reference point is :,Cont.,About arbitrary point,The arbitrary reference point is :,Cont.,則變換矩陣為:,Shear transformations,Dependence axis(依賴軸）: corresponding coordinate is remained Direction axis（方向軸）: corresponding coordinate is changed linearly Representations:,變換的一般表達(dá)

3、式是：,Shear transformations,Parameters: rotate axis, rotate angle 二維旋轉(zhuǎn)變換是三維空間中繞Z軸的旋轉(zhuǎn),記為：,Rotate transformation,Rotate about X axis,Equally with changing the coordinate system x,y,z to the coordinate system y,z,x.,Rotate about Y axis,Changing system x,y,z to system z,x,y,?:about arbitrary line,是關(guān)于某直線或平

4、面進(jìn)行的 關(guān)于某個軸進(jìn)行的反射變換等同于關(guān)于該軸做180度的旋轉(zhuǎn)變換 For instance: about Z axis,Reflect transformation（反射變換）,?:about arbitrary symmetry axis,Cont.,當(dāng)反射平面是坐標(biāo)平面時，等同于進(jìn)行左、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 For instance: about XOY plane,?About arbitrary symmetry plane,For instance: rotating about arbitrary line Overlapping arbitrar

5、y line with Z axis Resolving a series of problems Reflect about an arbitrary symmetry line Reflect about an arbitrary symmetry plane,Composition transformations,旋轉(zhuǎn)軸不與坐標(biāo)軸重合時變換的實(shí)現(xiàn)： 經(jīng)復(fù)合變換使旋轉(zhuǎn)軸與某坐標(biāo)軸重合 繞指定軸進(jìn)行旋轉(zhuǎn)變換 還原坐標(biāo)系,Rotate about arbitrary line,（1）translate P1 to overlap origin,不妨設(shè)P1P2為方向單位矢量，P2點(diǎn)坐標(biāo)為(a,b

6、,c),Cont.,P1,P2,Cont.,Cont.,(2)rotate about X axis to put the line on XOZ,X,Y,Z,Cont.,(2)rotate about X axis to put the line on XOZ,X,Y,Z,Cont.,(2)rotate about X axis to put the line on XOZ,X,Y,Z,Cont.,(2)rotate about X axis to put the line on XOZ,X,Y,Z,Cont.,(2)rotate about X axis to put the line o

7、n XOZ,X,Y,Z,Cont.,(2)rotate about X axis to put the line on XOZ,X,Y,Z,Cont.,(2)rotate about X axis to put the line on XOZ,X,Y,Z,Cont.,(2)rotate about X axis to put the line on XOZ,Then P2 is (a,0,d),Transformation matrix(變換矩陣）,Cont.,(3) Rotate about Y axis to overlap the line with Z axis,X,Y,Z,Cont.

8、,X,Y,Z,(4) Rotate about Z axis namely the line through ,Cont.,X,Y,Z,P1,P2,Cont.,(4)recover the coordinate system The final transformation is: R()=T1-1Rx-1(-)Ry-1() Rz()Ry()Rx()T1,Cont.,關(guān)于任意直線(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 out classroom,Exercise 4.11 Giv

9、en 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(對角線） (from( 0,0,0) to (1,1,1) in the counterclockwise direction when looking along the diagonal toward the origin.,Exercises out c

10、lassroom,Exercise 4.14 An object is to be scaled by a factor S in the direction whose direction cosines are (,).Derive the transformation matrix .,Two methods of transformation,Coordinate system fixed, Graphics changed Graphics fixed, Coordinate system changed,(1)坐標(biāo)系不變,圖形變換;,(2)圖形不變,坐標(biāo)系變換.,變換的兩種實(shí)現(xiàn)方法

11、:,Transforming coordinate system,Two means: Define the new coordinate system directly Define a vector in y direction of the new coordinate system,Cont.,1. Define a new system: composition of transformations,(1) translate: T（-x0，-y0） (2) rotate:R(-) (3) scale (4) composition of above transformations

12、(notice the sequence),Cont.,The matrix is:,Cont.,2. Define a vector in y direction of new system:,Y axis is:,X axis is:,Transformation is:,Contrast,VS.,Transform from an old coordinate system to another new coordinate system The new system is shown in the right figure:,Mode transformation,Cont.,Comp

13、osition of translation and rotation:,當(dāng)坐標(biāo)系使用不同的縮放時，還需定義縮放補(bǔ)償。,4.3 window-to-viewport transformation,World Domain(用戶域WD) 指程序員用來定義草圖的整個自然空間. World-coordinate system(用戶坐標(biāo)系WC). 世界坐標(biāo)系 右手直角坐標(biāo)系 Window(窗口區(qū)W) 在用戶坐標(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)域稱為窗口,Related concepts,Cont.,Screen Domain(屏幕域SD) 設(shè)備輸出圖形的最大區(qū)域,是有限的整數(shù)域. Viewport(視圖區(qū)V) 在顯示器坐標(biāo)系中規(guī)定的顯示圖形的區(qū)域稱為視（圖）區(qū). Screen coordinates（屏幕坐標(biāo)系） (normalized) device coordinates device coordinates: addressing by pixels NDC: -1，1-a，a,Window as a viewfinder,Cont.,視見變換將用戶坐標(biāo)系中窗口內(nèi)的圖形變換到顯示器中的視見區(qū)中產(chǎn)生顯示.,Window-to-Viewport transformation,Cont.,Cont.,transform matrix,窗口,Wxl,Wxr,Wyb