y z P 3 P T P1 P 2. Werner Purgathofer. b a
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1 Einführung in Viual Compuing Einführung in Viual Compuing in co T P 3 P co in T P P 2 co in Geomeric Tranformaion Geomeric Tranformaion W P h f Werner Purgahofer b a
2 Tranformaion in he Rendering Pipeline cene objec in objec pace objec capure/creaion i modeling viewing projecion vere age ( vere hader ranformed verice in clip pace cene in normalied device coordinae raer image in piel coordinae Werner Purgahofer 2 clipping + homogeniaion viewpor ranformaion hading raeriaion i piel age ( fragmen hader
3 Geomeric Tranformaion in he Rendering Pipeline cene objec in objec pace objec capure/creaion i modeling viewing projecion vere age ( vere hader ranformed verice in clip pace cene in normalied device coordinae raer image in piel coordinae Werner Purgahofer 3 clipping + homogeniaion viewpor ranformaion hading raeriaion i piel age ( fragmen hader
4 Baic Tranformaion: Tranlaion ranlaing a poin from poiion P o poiion P wih ranlaion vecor T P' P T P P T noaion: P, P, T Werner Purgahofer 4
5 Baic Tranformaion: Tranlaion rigid bod ranformaion objec ranformed b ranforming boundar poin P 3 T P 2 P Werner Purgahofer 5
6 Baic Tranformaion: Roaion roaion of an objec b an angle around d he pivo poin ( r, r r r r r Werner Purgahofer 6
7 Baic Tranformaion: Roaion poiive angle ccw roaion = r. co = r. in (, = r. co(+ = r. co. co r. in. in =. co.ini r = r. in(+ r (, = r. co. in r. in. co =. co. in =. in. co Werner Purgahofer 7
8 Baic Tranformaion: Roaion formulaion wih a ranformaion mari =. co. in =. in. co P R P wih R co in in i co RP co in in co co in in co Werner Purgahofer 8
9 Baic Tranformaion: Scaling =., =. P P = S P P S eample: a line caled uing = =.33 i reduced in ie and moved cloer o he coordinae origin Werner Purgahofer 9
10 Baic Tranformaion: Scaling uniform caling: differenial caling: fied poin: ( f, f Werner Purgahofer
11 Tranformaion Marice caling roaion co in in co mirroring ranlaion ( = (+,, + (' '...? Werner Purgahofer
12 Homogeneou Coordinae ( inead of ue h h ver ofen h=, i.e. h wih = h /h, = h /h in hi wa all ranformaion can be formulaed in mari form Werner Purgahofer 2
13 Homogeneou Coordinae (2 Homogeneou Coordinae (2 ranlaion noaion: ranlaion P T P, ( P T P, ( roaion in co co in P R P ( li caling P S P ( P S P, ( Werner Purgahofer 3
14 Invere Marice ( ranlaion T (, T (, roaion R ( R( caling S (, S (/,/ Werner Purgahofer 4
15 Compoie Tranformaion ( n ranformaion are applied afer each oher on a poin P,, hee ranformaion are repreened b marice M, M 2,..., M n., 2,, n P = M PP P = M 2 P... P (n = M P n P (n- horer: P (n = (M n...(m 2 (M P... Werner Purgahofer 5
16 Compoie Tranformaion (2 P (n = (M n... (M 2 (M P... mari muliplicaion are aociaive: (M M 2 M 3 = M (M 2 M 3 (bu no commuaive: M MM 2 M 2 MM Werner Purgahofer 6
17 Tranformaion are no commuaive! Revering he order in which a equence of ranformaion i performed ma affec he ranformed poiion of an objec! in (a, an objec i fir ranlaed, hen roaed. in (b, he objec i roaed fir, hen ranlaed. (a (b Werner Purgahofer 7
18 Compoie Tranformaion (2 P (n = (M n... (M 2 (M P... mari muliplicaion are aociaive: (M M 2 M 3 = M (M 2 M 3 (bu no commuaive: M MM 2 M 2 MM herefore he oal ranformaion can alo be wrien a: P (n = ( M n... M 2 M P conan for whole image, objec, ec.!!! Werner Purgahofer 8
19 Compoie Tranformaion (3 Compoie Tranformaion (3 imple compoie ranformaion compoie ranlaion compoie ranlaion ( ( ( T T T, (, (, ( T T T compoie roaion ( ( ( 2 2 R R R compoie caling ( ( ( 2 2 compoie caling ( ( ( S S S, (, (, ( S S S Werner Purgahofer 9
20 Compoie Tranformaion (4 general pivo poin roaion T (, R ( T (, R (,, r r r r r r original ranlaion of poiion and objec o ha pivo poin pivo poin i a origin roaion ranlaion abou o ha he origin pivo poin i reurned Werner Purgahofer 2
21 Compoie Tranformaion (5 T general fied poin caling, S (, T (, S (,,, f f f f f ( f ( f, f ( f, f original poiion and fied poin ranlae objec o ha fied poin i a origin cale objec wih repec o origin ranlae o ha he fied poin i reurned Werner Purgahofer 2
22 Compoie Tranformaion (6 general caling direcion R ( S(, 2 R(( ( 2 (,2 2 (2,2 2 (, (, 2 original afer 45 afer (,2 afer - 45 poiion roaion caling roaion back Werner Purgahofer 22
23 Eample ranlae b (3,4, hen roae b 45 and hen cale up b facor 2 in direcion. M = T(3,4 = 3 4 co 45 in 45 in 45 co M 2 = R(45 = 3. M M3 = S(2, = 2 Werner Purgahofer 23 M = M M M 3 2
24 Eample ranlae b (3,4, hen roae b 45 and hen cale up b facor 2 in direcion M = M 3 M 2 M = 2 co 45 in 45 = in 45 co 45 2 = Werner Purgahofer co 45 in i = 3co 45 4i 4in 45 in 45 co 45 3i 3in co 45 = = 2co 45 in 45 in i 45 co 45 6co 45 8i 8in 45 3i 3in co 45
25 Reflecion abou ai: i abou ai: i Rf = Rf = Werner Purgahofer 25
26 Eample reflecion abou he ai wih angle = Werner Purgahofer 26
27 Eample reflecion abou he ai wih angle. roaion b 2. mirroring abou ai 3. roaion b = + + Werner Purgahofer 27
28 Eample reflecion abou he ai wih angle co( in(. M = R( = in( co( 2. M 2 = S(, = 3. M 3 = R( =. 2. co in co in 3. P =M 3 ( M 2 ( M P= (M 3 M 2 M P Werner Purgahofer 28
29 Eample reflecion abou he ai wih angle M 3 M 2 M = co in in co co( in(( in( co( = = co in in co co in in co = = co 2 in 2 2inco co2 in2 = 2inco in 2 co 2 = in2 co2 Werner Purgahofer 29
30 Oher Tranformaion: Reflecion abou a Poin reflecion abou origin Rf O (=R(8 ( = Werner Purgahofer 3
31 Reflecion wih Repec o a General Line reflecion wih repec o he line =m+b m+b T(,b R( S(, R( T(,b m = an( Werner Purgahofer 3
32 Oher Tranformaion: Shear ( direcion hear along ai reference line = (, (, h (h, (h +, (, (, (, (, Werner Purgahofer 32
33 Oher Tranformaion: Shear (2 general direcion hear along ai reference line = ref h h ref h Werner Purgahofer 33 ref
34 Oher Tranformaion: Shear (3 general direcion hear along ai reference line = ref h h ref ref h Werner Purgahofer 34
35 Tranformaion beween Coordinae Sem M R ( T (,, A Careian em poiioned a (, wih orienaion in an Careian em Poiion of he reference frame afer ranlaing he origin of he em o he coordinae origin of he em Werner Purgahofer 35
36 Affine Tranformaion a a b a a b collinear poin on a line a on a line parallel line parallel line raio of diance along a line are preerved finie poin finie poin an affine ranformaion i a combinaion of ranlaion, roaion, caling, (reflecion, hear ranlaion, roaion, reflecion onl: angle, lengh preerving Werner Purgahofer 36
37 3D Tranformaion all concep can be eended o 3D in a raigh forward wa + projecion 3D 2D Werner Purgahofer 37
38 3D Tranlaion ( 3D Tranlaion ( ranlaion vecor (,,,, P T P ( P T P,, ( Werner Purgahofer 38
39 3D Tranlaion (2 3D Tranlaion (2 objec ranlaed b ranlaing boundar poin invere:,, (,, ( T T ( ( Werner Purgahofer 39
40 3D Roaion: Angle Orienaion roaion ai poiive angle counerclockwie roaion Werner Purgahofer 4
41 3D Roaion: Coordinae Ae ( ai co in in co co in in co P R ( P Werner Purgahofer 4
42 3D Roaion: Coordinae Ae ( ai co in in co P R ( P Werner Purgahofer 42
43 3D Roaion: Coordinae Ae ( ai co in in co P R ( P Werner Purgahofer 43
44 3D Roaion: Ai Parallel o Ai original objec poiion. ranlae roaion ai ono ai: T 2. roae objec hrough angle q 3. ranlae R ( T R ( T roaion ai o original ii poiion: T Werner Purgahofer 44
45 3D Roaion around Arbirar Ai an ai of roaion (dahed line defined wih poin P and P 2. The direcion of he uni ai vecor u deermine he roaion direcion. P 2 u P P 2 ( a, b, c P P 2 u P Werner Purgahofer 45
46 3D Roaion around Arbirar Ai u P. ranlae 2. roae u iniial poiion P o origin ono ai P u 3. roae objec 4. roae ai o 5. ranlae ai o around ai original orienaion original poiion Werner Werner Purgahofer Purgahofer 46 46
47 3D Roaion around Arbirar Ai 3D Roaion around Arbirar Ai ep : ranlaion T(,,,, ( T u u P Werner Purgahofer 47
48 3D Roaion around Arbirar Ai u P. ranlae 2. roae u iniial poiion P o origin ono ai P u 3. roae objec 4. roae ai o 5. ranlae ai o around ai original orienaion original poiion Werner Werner Purgahofer Purgahofer 48 48
49 u 3D Roaion around Arbirar Ai ep 2: roaion o ha u coincide wih ai (done wih wo roaion R (a: u plane R (b: u ai u 2a: 2b: P Werner Purgahofer 49
50 u 3D Roaion around Arbirar Ai ep 2a: u d b 2 c 2 u ( a, b, c u (,b,c bc u u co = c/d R ( c/ d b/ d b/ d c/ d b d c P u = (a,,d Werner Purgahofer 5
51 u 3D Roaion around Arbirar Ai ep 2b: u (,b,c co = d d a d in = -a R ( d a d u u (a,, u = (,, d a P u = (a,,d Werner Purgahofer 5
52 3D Roaion around Arbirar Ai u P. ranlae 2. roae u iniial poiion P o origin ono ai P u 3. roae objec 4. roae ai o 5. ranlae ai o around ai original orienaion original poiion Werner Werner Purgahofer Purgahofer 52 52
53 u 3D Roaion around Arbirar Ai ep 3: u aligned wih ai roaion around ai P Werner Purgahofer 53
54 3D Roaion around Arbirar Ai u P. ranlae 2. roae u iniial poiion P o origin ono ai P u 3. roae objec 4. roae ai o 5. ranlae ai o around ai original orienaion original poiion Werner Werner Purgahofer Purgahofer 54 54
55 u 3D Roaion around Arbirar Ai ep 4: undo roaion of ep 2 ep 5: undo ranlaion of ep R(θ = T - (P R R - (α R R - (β RR (θ RR (β R(α T(P T(P ep: 5 4a 4b 3 2b 2a invere of roaion: R ( T R ( R ( P Werner Purgahofer 55
56 3D Scaling wih repec o Origin 3D Scaling wih repec o Origin doubling he ie of an objec alo doubling he ie of an objec alo move he objec farher from he move he objec farher from he origin origin P S P Werner Purgahofer 56
57 3D Scaling wih oher Fied Poin T( F, F, F F S(,,, T( F F,, F, F ( F, F, F Werner Purgahofer 57
58 3D Scaling wih oher Fied Poin T( F, F, F F S(,,, T( F F,, F, F ( F, F, F (,, Werner Purgahofer 58
59 3D Scaling wih oher Fied Poin T( F, F, F F S(,,, T( F F,, F, F ( F, F, F (,, Werner Purgahofer 59
60 3D Scaling wih oher Fied Poin T( F, F, F F S(,,, T( F F,, F, F ( F, F, F (,, Werner Purgahofer 6
61 3D Scaling wih oher Fied Poin T( F, F, F S(,, T( F, F, F ( ( ( F ( F F Werner Purgahofer 6
62 3D Reflecion reflecion wih repec o poin line (8 roaion plane, e.g., plane: l RF P'(a,b,-c P(a,b,c RF reflecion relaive o plane Werner Purgahofer 62
63 3D Shear eample: hear relaive o -ai wih a=b= a b SH a b Werner Purgahofer 63
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