计算机图形学3D变换.ppt

1、4.3 3D transformations,Translate(平移) transformations Rotate(旋转) transformations Scale(缩放) transformations Reflect(反射) transformations Shear(错切) transformations Composition(复合) of 2D transformations,与二维平移变换类似地使用齐次坐标表示为：,记为：,其中,Translate transformation,Translate transformation,Remarked:,Whereas:,Trans

2、late,记为：,Scale transformation,About origin,Cont.,About arbitrary point,The arbitrary reference point is :,Consists of:translate, scale about origin, inverse translate,Cont.,则变换矩阵为:,About arbitrary point,The arbitrary reference point is :,Cont.,About arbitrary point,The arbitrary reference point is :

3、,Cont.,则变换矩阵为:,Parameters: rotate axis, rotate angle 二维旋转变换是三维空间中绕Z轴的旋转,记为：,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,是关于某直线或平面进行的 关于某个轴进

4、行的反射变换等同于关于该轴做180度的旋转变换 For instance: about Z axis,Reflect transformation,?:about arbitrary symmetry axis,Cont.,当反射平面是坐标平面时，等同于进行左、右手坐标系的互换，相应变换矩阵是把第三维坐标值取反 For instance: about XOY plane,?About arbitrary symmetry plane,Cont.,关于任意直线(或平面)的反射可以分解为平移、旋转（使得指定的反射直线或平面与某坐标轴或平面重合）和关于坐标直线(或平面)的反射。,Shear tran

5、sformations,Dependence axis: corresponding coordinate is remained Direction axis: corresponding coordinate is changed linearly Representations:,Cont. Representation,变换的一般表达式是：,For instance: rotating about arbitrary line Overlapping arbitrary line with Z axis Resolving a series of problems Reflect ab

6、out an arbitrary symmetry line Reflect about an arbitrary symmetry plane,Composition transformations,旋转轴不与坐标轴重合时变换的实现： 经复合变换使旋转轴与某坐标轴重合 绕指定轴进行旋转变换 还原坐标系,Rotate about arbitrary line,（1）translate P1 to overlap origin,不妨设P1P2为方向矢量，P2点为(a,b,c),Cont.,P1,P2,Cont.,Cont.,(2)rotate about X axis to put the li

7、ne 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 on XOZ,X,Y,Z,Cont.,(2)rotate about X axis to put the line on XOZ,

8、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,Cont.,(3) Rotate about Y axis to overlap the line with Z axis,X,Y,Z,Cont.,(4) Rotate about Z axis namely the line through ,Cont.,Cont.,(4)recov

9、er the coordinate system The final transformation is: R()=T1-1Rx (-)Ry () Rz()Ry(-)Rx()T1,Two methods of transformation,Coordinate system fixed, Graphics changed Graphics fixed, Coordinate system changed new coordinate system is saw as a graphics and transformed to overlap with the original coordina

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