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No problem; right column is translation vector8[Z0[  [Translation Txyz(a,b,c)$ *        JTransforming Point(  Can represent affine transform as block matrix multiplication Transform point P = (x, 1), get So y is translation part, what does A represent?nN   |ATransforming Vectors(  uRepresent vector with w = 0 (x, y, z) => (x, y, z, 0) Can also write as v => (v, 0) Think of as point at infinityZ4     KTransforming Vectors(  Apply affine transform to vector, get Applies linear transform A to vector So linear transforms are subset of affine transforms Note no translation: makes sense, vectors have no position!(ZA|,cGNOther Affine Transformations(*      Remaining affine transformations are also linear transformations Can be applied to vectors as well Notice that the top-left 3x3 block in the translation example was the identity? Just place the linear transform in the upper-left 3x3 block, as follows&  \Scaling Sxyz(a,b,c)$ *       (  ]>Rotating About the x-axis Rx(q)6     (  ^@Rotating About the y-axis Ry(q)6!*         _@Rotation About the z-axis Rz(q)6!*         `Rotation Matrices  6Have some useful properties Are orthogonal, i.e., R-1 = RT Column vectors are transformed axis of old coordinate system We ll see how we can use this laterp  a  a Concatenation of Transformations!  Transforms concatenated into a single matrix by multiplying them together, in the order they are to occur Can then use single matrix to perform transforms on many points@ 4?   Concatenation   Can perform a series of transformations v = M1 v v  = M2 v w = M3 v  Substitute to get w = M3 M2 v w = M3 M2 M1 v ('&(           Concatenation  Can combine series into one matrix that performs all of them M = M3 M2 M1 w = M v Example: Euler rotations>>   *c       b Euler Angles    ZCommon representation for 3D orientation (common best) Three Euler angles: Yaw = shaking your head,  no Pitch = nodding your head,  yes Roll = cocking your head sidewaysjMa1*?    j  cThe Euler Transform*       TEach angle corresponds to a rotation about an axis (dependent on coordinate system) TUU  dThe Euler Transform*       6Looking down the x-axis: Yaw = Rz Pitch = Ry Roll = Rx7   F           eThe Euler Transform*       For this coordinate system, the Euler Transform E E(y,p,r) = Rx(r) Ry(p) Rz(y) Again, E is orthogonal (it s a pure rotation matrix)F3 50   ,b     !        :  fConcatentation of Transforms    Can concatenate any sequence of scale, rotation, translate Matrix products are non-commutative, so the order matters Translating the point (0, 0, 0) and then rotating will not yield the same point as rotating (0, 0, 0) then translating (rotation of (0, 0, 0) does nothing)fZ%Z =   VConcatentation of Transforms    Similarly, scaling and rotation are non-commutative Example: rotate (1 0 0) by 90 around z, scale by (sx, sy, sz), get (0 sy 0) scale (1 0 0) by (sx, sy, sz), rotate (1 0 0) by 90 around z, get (0 sx 0)>ZZ>         k                             *      gTransformation Order  UTo perform scale, then rotation, then translation (generally the order desired), use V  hMatrix Breakdown  lRotation, scale in upper left 3x3 Translation rightmost column Does rotation/scale first, then translatem  i Better Format  Problem: want to Translate object in space and change rotation and scale arbitrarily Handle rotation/scale separately Can t do easily with 4x4 matrix format Involves SVD, Polar decomposition Messy, but we can do better using& 6elel  BRigid Body Transforms  Any sequence of translation and rotation transformations Not scale or shear Object shape is not affected (preserves angles and lengths) Usually include uniform scale despite this  j Better Format  Scale  one uniform scale factor s Rotation  3x3 matrix R Translation  single vector tXY!Y  k Better Format  BWant to concatenate transforms T1, T2 in this form, or Do this byRC  C  l Better Format  Advantages Clear what each part does Easier to change individual elements Concat: 39 mult 27 add vs. 48 mult 32 add Disadvantages Eventually have to convert to 4x4 matrix anyway for renderer/video cardL iH iH~J            J       m Better Format  Matrix conversion  CInverting Rigid Body Xforms*      <We can easily invert our rigid body transforms symbolically:=  DInverting Rigid Body Xforms (2) (*      HIn fact, the result itself may be written as a rigid body transform: H  o References   Rogers, F. 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