《计算机图形学教学资料》(1)ppt课件

1、Chapter 4 Geometrical Transformations1.Vector(矢量2.Matrix矩阵Mathematics Preparation for transformation1两个矢量之和：矢量运算(Vector)2两个矢量的点积：21212121zzyyxxVV),(21212121zzyyxxVV3矢量的长度：2111111121111)()(zzyyxxVVV矢量运算4两个矢量的叉积：),(),(12211221122122112211221122211121yxyxxzxzzyzyyxyxxzxzzyzyzyxzyxkjiVVNoImagemnmmnnaaaa

2、aaaaaA212221111211aij是矩阵中第i行第j上的元素。上述矩阵记为A，或AmXn，或(aij) mXn。矩阵(matrix)Cont.ndefinition：mnmmnnmnaaaaaaaaaA212222111211m=n时：A称为方阵m=1时：退化为行向量(vector)n=1时：退化为列向量(vector) 时：称为A=Bijijba n nm mn na aa aa aA A11211 Cont.n单位矩阵：100010001nnIn零矩阵：0000000000mnICont.n矩阵加法：mnmnmnCBAijijijbac其中：n矩阵加法满足的运算律：n结合律n交换律

3、nA+0=ACont.n矩阵数乘mnmmnnmnijmnkakakakakakakakakakakA212222111211)(Cont.n矩阵乘法：mpnpmnCBAnlljilijbac1其中：n矩阵乘法满足的运算律：n结合律n分配律nAI=IA=An不满足交换律！Cont.n矩阵转置nmmnnnmmTmnBaaaaaaaaaA212221212111n矩阵转置运算律：TTTTTTTTTTABBAkAkABABAAA)()4()()3()()2()() 1 (Cont.n矩阵行列式n非奇特矩阵n逆矩阵：非奇特矩阵具有逆矩阵IAAAA11假设把n维空间坐标叫做普通坐标,那么相应的n+1维空间

4、坐标叫做齐次坐标. 齐次坐标技术:用n+1维向量表示n维向量的技术ordinary coordinatehomogeneous coordinate(x，y)(hx，hy，h)其中，h是一个恣意的非零标量齐次坐标(homogeneous coordinate)齐次坐标(homogeneous coordinate)0),(),(0),(),(hhhzhyhxzyxhhhyhxyx1husually那么 (x，y)的齐次坐标为(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代数式矩阵式3D spacen右手直角坐标系XYZnPoints: zyxPAlgebraic representation of 3D transformations100013433323124232221

6、14131211zyxazayaxazazayaxayazayaxax),( :),( :zyxafterzyxbefore；Matrix representation1144434241343332312423222114131211zyxaaaaaaaaaaaaaaaazyxnIntroduce homogeneous coordinate0 0 0 1变换的两种实现方法: (1)坐标系不变,图形变换; (2)图形不变,坐标系变换.Transformation method4.1 2D transformations nTranslate(平移) transformationsnRotat

7、e(旋转) transformationsnScale(缩放) transformationsnReflect(反射) transformationsnShear(错切) transformationsnComposition(复合) of 2D transformations 在二维坐标系中,将点P(x,y)在x、y轴方向分别平移tx、ty，得到点P(x,y)，那么P点与P点的坐标关系为： yxtyytxx变换矩阵:T=yxtt矢量方式为：P=P+TTranslate transformations(平移变换)Before translationAfter translation(4,5)(

8、7,5)xy(7,1)(10,1)xy yxttyxyxCont.用齐次坐标表示平移变换过程：.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.变换矩阵：1000cossin0sincos)(RPRP)( 那么变换公式为：vPositive : anti-clockwise rotate(逆时针) vNegative : clockwise rotate(顺时针)逆变换逆变换:)(RCont.nRotating an

10、 object图元的旋转变换n旋转变换是刚体变换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)固定某个点的旋转变换?Scaling tran

11、sformations1ysyxsxyx变换矩阵:1000000yxssS变换公式:1),(1yxssSyxyx缩放变换是指对点的缩放变换是指对点的X，Y坐标值进展缩放。坐标值进展缩放。Sx , Sy 称为缩放系数，可取任何正数; S称为缩放矩阵。缩放变换可使物体产生重定位，如右图所示，缩放比例不同，定位间隔也不同。当缩放系数大于1时，物体被放大，否那么减少；当SxSy时，物体发生等比变换，否那么发生差值缩放，产生变形。Scaling transformations2P0P1OP1P0Scaling an objectnPolygon多边形nScaling the vertexes and t

12、hen re-defining the polygonncircle圆中心对称图形nScaling the radiusnPrimitives defined by some parameters给定定义参数的图形nScaling the parameters and re-defining the primitivesyyxx相对X轴对称：Symmetry transformations变换矩阵:100010001xSY变换公式:11yxSYyxx对称变换是产生物体镜象的一种变换.Cont.相对Y轴对称:11000100011yxyxyyxxCont.11000100011yxyx相对原点对

13、称:yyxxCont.11000010101yxyx11000010101yxyx相对y=x直线对称:相对y=-x直线对称:xyyxxyyx错切变换：坚持图形上各点的某一坐标值不变，错切变换：坚持图形上各点的某一坐标值不变，而另一坐标值关于该坐标值呈线性变化。而另一坐标值关于该坐标值呈线性变化。坐标坚持不变的坐标轴称为倚赖轴，其它坐标坐标坚持不变的坐标轴称为倚赖轴，其它坐标轴称为方向轴。轴称为方向轴。(1)以以y轴为依赖轴的错切变换轴为依赖轴的错切变换(沿沿x方向的错切方向的错切 即即y不变，不变，x的值随的值随y的值而线性变化的值而线性变化yyyshxxxShx=tan Shear tran

14、sformationsyxshyxxyShy=tan (2)以x轴为依赖轴的错切变换Cont.Cont.nTwo kinds:nShear along x axisnShear along y axisnGeneral representation of shear:11bxydyxyxb=0 or d=011000101yxbd变换合成n根据矩阵运算的性质，可推出二维根本变换的如下性质：n平移变换和旋转变换具有可加性n缩放变换具有可乘性),(),(),(21211122yyxxyxyxttttTttTttT)()()(2112RRR),(),(),(21211122yyxxyxyxssssS

15、ssSssSRotate transformation about arbitrary point),()(),(0000yxyxTRTT总),(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总TRSY1R1T作业：关于恣意直线的对称变换直线为 y= x+b写出 T总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经过变换后的两个端点可定义变换后的直线Exercises nExe