FCC Graficación CAD/CAM3D TransformationsContents1. Translation2. Scaling3. Rotation4. Other Transformations1. TranslationxyzTxTzTy(x,y,z))',','(zyx110001000100011'''zyxTTTzyxTzzTyyTxxzyxzyxTPP'SPP'2. Scaling110000000000001'''zyxSSSzyxSzzSyySxxzyxzyxxyz(x,y,z) ) ' , ' , ' ( z y x ∴scaling uniform-non Otherwise,scaling UniformzyxSSSxyz),,(FFFzyxxyzxyzxyz.1000100010001 and,1000000000000,1000100010001where]][][['11FFFzyxFFFxxxTSSSSxxxTPTSTPRelative Scaling3. Rotation2D Rotation)','(yx),(yxqrr ) ' , ' ( y x ) , ( y x qrr),(RRyxRotation about z-axis is implicit !!!)',','('zyxP),,(zyxPqrrxyzzzyxyyxx'cossin'sincos'∴11000010000cossin00sincos1'''zyxzyxPRPZ'By symmetry,xyz10000cossin00sincos00001where,'XXRPRPxyz10000cos0sin00100sin0coswhere,'YYRPRPcossin'sincos''zyzzyyxxcossin''sincos'zxzyyzxxxyzu(x1, y1, z1)(x2, y2, z2)Rotation about an arbitrary axisBasic idea1. Translatethe object so thatthe rotation axis passes through the origin.2. Rotatethe object so thatthe rotation axis coincides with one of the coordinate axis.3. Perform the specified rotation.4. R-15. T-1TR1,,,,),,(),,(222222121212121212cbaVzcVybVxazyxVzzzyyyxxxcbaVVuzzyyxxVzyxStep 1: Translationxyzu = (a, b, c)(x1, y1, z1)(x2, y2, z2)u1000100010001111zyxTStep 2: Aligning uwith z-axisxyzu = (a, b, c)a(0, b, c) = u'(0, 0, 1) = uzxyzu''= (a, 0, d)da100000000001)(dcdbdbdcRx)(1000'sinsin|||'|'22bucbuuuuucbuuuuuuxzyxzxzxz uxdcuuuucbbbcbzz|||'|'cossinsin2222 dxyzu''= (a, 0, d)dauy(0, 0, 1) = uzbxyzaaudauuuuuuuuuuuyzyxzyzyzsin)(1000''sinsin|||''|''∴100000001000)(daadRyduuuuzz|||''|''cos≡1≡1111)()()()(yxxyRRRRRR∴In Step 2,for later useStep 3: Rotate about z-axis by a given anglexyzq1000010000cossin00sincos)(zRStep 4: [R-1] = [Rx(a)-1][Ry(b)-1]Step 5: [T-1]∴In summary,PTRRRRRTPxyzyx)()()()()('1114. Concatenation Thesuccessiveapplicationofanumberoftransformationscanbeachievedwithasingletransformationmatrix,theconcatenationofthesequence.SupposetwotransformationsT1andT2aretobeappliedsuccessively.ThesameeffectcanbeachievedbytheapplicationofasingletransformationT3,whichissimplytheproductofthematricesT1andT2.Thatis:The point (x, y, z) is transformed into (x’, y’, z’) by T1:[ x’ y’ z’ 1 ] = [ x y z 1 ] T1(1)The point (x’’, y’’, z’’) is generated by applying T2:[ x’’ y’’ z’’ 1 ] = [ x’ y’ z’ 1 ] T2(2)Substituting (1) in (2) gives:[ x’’ y’’ z’’ 1 ] = ([ x y z 1 ] T1)T2 = [ x y z 1 ] (T1T2)Theorderofapplicationofthetransformationsmustbepreservedwhenthetransformationmatricesaremultipliedtogether.