Computer Graphics. Geometric. Transformations. by Brian Wyvill University of Calgary. Lecture 2 Geometric. Transformations. Lecture 2 Geometric

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1 Lecture 2 Geometric Transformations Computer Graphics Lecture 2 Geometric Transformations b Brian Wvill Universit of Calgar ENEL/CPSC. Lecture 2 Geometric Transformations Lecture 2 Geometric Transformations Lecture 2 Geometric Transformations Geometric Transformations Lecture 2 Geometric Transformations Lecture 2 Geometric Transformations

2 2D Transformations TRANSLATION 2,3 2, 0,0 4,0 0,0 Object Description: Object Description: new = + d new = + d Using Vector Notation new P = P new = T = so P new = P + T Y new d d ENEL/CPSC 2. F&VD.2 p68

3 2D Transformations SCALING 2,3 2, 0,0 4,0 0,0 Object Description: Scale b Object Description: new = * S new = * S Using Matrices S is the scaling matri new new S 0 = * or P new = S * P 0 S ENEL/CPSC 3.

4 2D Transformations ROTATION 2,3 2, 0,0 4,0 0,0 Object Description: Rotate b 5 degrees Object Description: new = * cosθ * sinθ new = * sinθ + * cosθ Using Matrices: R is the rotation matri new new cosθ sinθ = * or P new = R * P sinθ cosθ ENEL/CPSC 4.

5 Rotation Positive angles are measured anti clockwise from towards. P new ( new new ) P(,) r r θ φ 0,0 new = * cosθ * sinθ new = * sinθ + * cosθ (2.) From the diagram: = r cosφ, = r sinφ (2.2) new = r cos(θ + φ) = r cosφ cosθ r sinφ sinθ new = r sin(θ + φ) = r cosφ sinθ + r sinφ cosθ Substituting for and from 2.2 we get 2. ENEL/CPSC 5.

6 Homogeneous Coordinates We have : P new = P + T Translation P new = S * P Scaling P new = R * P Rotation Note the addition for Translation. B epressing points as homogeneous coordinates all three affine transformations can be treated the same. In homogeneous coordinates becomes:,,w represents the same point as,,w if one is a multiple of the other. Thus (2,3,6) is the same as (4,6,2) since 2/6 = 4/2 and 3/6 = 6/2 (W!= 0) In particular: (,,W) is the same point as (/W, /W, ) (W!= 0) Dividing the homogeneous point b W gives the Cartesian point (/W, /W) Note the dimension change. W ENEL/CPSC 6.

7 Homogeneous Representation of 2D transforms W 3 space Line representing all the triples of the form (t, t, tw) t!=0 each 2D homogeneous point represents a line of points in 3 space. W= plane The triples found b dividing b W: (,,) represent points in 2 space. These homogenized points form a plane in (,,W) space where W= ENEL/CPSC 7.

8 Homogeneous Representation of 2D transforms continued Points are now 3 element column vectors so we need 33 matrices to represent tranformations. The translation equation becomes: new new 0 d = 0 d * 0 0 CAUTION!! M earlier notes (some are still unconverted) and some tet books use row vectors and premultilication rather than postmultiplication. (New convention conforms to current tet book). Matrices can be transposed to go from one convention to the other: (P * M) T = M T * P T ENEL/CPSC 8.

9 Column Vector vs. Row Vector Column vectors take less space on the page in book format. However for the assignment premultipling matrics as the are read from the user is easier than stacking the matrics. Use whichever convention ou feel most comfortable with but be consistent. Row Vector Representation e.g. rotation around arbitrar point: 0 0 cosθ sinθ new new = 0 0 * sinθ cosθ 0 * 0 0 * d d 0 0 d d Column Vector Representation e.g. rotation around arbitrar point: new new 0 d cosθ sinθ 0 0 d = 0 d * sinθ cosθ 0 * 0 d * ENEL/CPSC 9.

10 Homogeneous Representation of 2D transforms continued new new 0 d = 0 d * 0 0 Translation new new S 0 0 = 0 S 0 * 0 0 Scaling new new cosθ sinθ 0 = sinθ cosθ 0 * 0 0 Rotation ENEL/CPSC 0.

11 Composing Geometric Transformations 2,3 2, 0,0 4,0 0,0 0,0 E.g. Translate b (0,) then translate b (,) First translation T0 = Net translation T = P = T0 * P P = T * P so P = T * T0 * P Matri product T * T0 is ENEL/CPSC.

12 Composing Geometric Transformations continued Matri product also known as concatanation, compounding or composition of matrices. Consider a set of n= points P representing the following piano mesh. The points are scaled b matri S and rotated b matri R and translated b matri T. Number of vector matri multiplies = n*s followed b n*r then n*t = 3n Alternativel M = R*S*T and then n*m matri vector calculations = n ENEL/CPSC 2.

13 Order of Concatanation 2 2 0,0 0,0 Start with a unit square at the origin. Rotatate about the origin translate to 3,2 Translate to 3,2 rotate Rotate about the origin 2 2 0,0 Scale about the origin b 2, translate to 3,2 0,0 Translate to 2, Scale about the origin b 2, ENEL/CPSC 3.

14 Order of Concatanation 0,0 0,0 Rotate b 45, scale b 2, translate to 2,3 Scale b 2, translate to 2,3 rotate b 45 0,0 Rotate b 45, scale 2,, rotate b 45. translate to 2,3 0,0 The square starts off at 2,3 What transformations are applied to produce the rotated version above? ENEL/CPSC 4.

15 Vector Librar A vector librar eists in ~blob/453/src/v3d_lib Tpes tpedef struct vt { double,,z; } VTX, *REFVTX; tpedef struct wvect { double,,z,w; } WVTX, *REFWVTX; tpedef struct { VTX pnt, normal; double d; } PLANE, *REFPLANE; tpedef struct { PLANE pln; int num; REFVTX pv; } POLY, *REFPOLY; tpedef double v3dmatrix[6], *REFv3dMATRIX; Functions returning a vector VTX v3dadd(vtx v, VTX v2); /* a+b */ VTX v3dsub(vtx v, VTX v2); /* vector from v to v2 */ VTX v3ddiff(vtx v,vtx v2); VTX v3dunit(vtx v); VTX v3dmul(vtx vec, double m); VTX v3ddiv(vtx vec, double m); VTX v3dcross( VTX v, VTX v2); VTX v3dnegate( VTX v); VTX v3dinterp (double t, VTX v, VTX v2); VTX v3dzero(); VTX v3dpointonplane(vtx p, REFPLANE pln); ENEL/CPSC 5.

16 Vector Librar continued Functions returning an int int nearzero(double ); int v3dnearzero(vtx v); int v3dsign(double ); int v3dsamedir(vtx a, VTX b); int v3dequal(vtx a, VTX b); int v3dcolinear(vtx a, VTX b, VTX c); Functions returning a double double v3dacos(double a); double v3dsigup( double t); double v3dlength( VTX v); double v3dsigdown( double t); double v3ddot( VTX v, VTX v2); double v3dsqlength(vtx v); /* returns square of length */ double v3dsqdist(vtx p, VTX p2); double v3dangle( VTX v, VTX v2); double v3dsangle( VTX v, VTX v2, VTX v3); double v3ddist( VTX p, VTX p2); double v3dbell( double t); double v3ddeg( double a); ENEL/CPSC 6.

17 Vector Librar continued Matri Functions void matmake(int desc, REFVTX value, double rotation, REFv3dMATRIX matri); void matassign(refv3dmatrix a, REFv3dMATRIX b); WVTX hmatpointmult(refv3dmatrix matri, WVTX p); VTX matpointmult( REFv3dMATRIX matri, VTX p); void matclear(refv3dmatrix matri); void matprint(file *outfile, char *buf, REFv3dMATRIX mat); REFv3dMATRIX matcreate(); REFv3dMATRIX matidentit( register REFv3dMATRIX matri); REFv3dMATRIX matcop( register REFv3dMATRIX m); REFv3dMATRIX matnth(refv3dmatrix mat, int n); REFv3dMATRIX matpremult(refv3dmatrix a, REFv3dMATRIX b, REFv3dMATRIX c); Print Functions /* print routines n= no newline f = full precision */ void v3dpvec( FILE *fp, char *buf, VTX v); void v3dpvecn(file *fp, char *buf, VTX v); void v3dpfvec( FILE *fp, char *buf, VTX v); void v3dpfvecn(file *fp, char *buf, VTX v); void v3dpplane(file *fp, char *buf, REFPLANE p); void v3dppol(file *fp, char *buf, REFPOLY p); void v3d2gl(vtx v, double *gv); /* convert to gl double [3] */ ENEL/CPSC 7.