AML7 CAD LECURE 5 D RANSFORMAIONS Con. Sequene of operons, Mr ulplon, onenon, onon of operons pes of rnsforon Affne Mp: A p φ h ps E 3 no self s lle n ffne Mp f leves renr onons nvrn. 3 If β j j,, j E hen, 3 φ β φ φ, φ E j j j Mos of he rnsforons h re use o poson or sle n oje n CAD re ffne ps. he ne ws gven L. Euler n sue ssell A. Mous Euln Mps: Rg o oons lke roon n rnslon where lenghs n ngles re unhnge re lle Euln ps. hs s spel se of ffne ps
rnsforon Groups n Seres Affne rnsforons re lssfe Fel Klen s follows Slr Groups: Rg oon n slng Eg: Congruen, slr Ser Groups:Roon, Refleon, rnslon. No rnslon n no refleon Crle group Eg: ger wheel. No rnslon n les one refleon Dherl group 3. rnslon long one reon. rnslon long ore hn one reon Hoogeneous oornes of veres A pon n hoogeneous oornes,, h, h, orrespons o he -D vere /h, /h n Cre oornes. Coneve h he Cre oornes es les on he plne of h. he nerseon of he plne n he lne onneng he orgn n,, h gves he orresponng Cre oornes. w,, h /h, /h, h h
For eple, oh he pons 6, 9, 3 n, 6, n he hoogeneous oornes orrespons o, 3 n he Cre oornes. Conversel, he pon, of he Cre orrespons o,,,,, or 6, 3, 3 of he hoogeneous sse h 6, 3, 3,,,, h h Roon Conser he followng fgure where poson veor p, whh kes n ngle φ o -s fer rnsforon o p, kes n ngle φ egrees. P, P [ ] [rosφ rφ] P [ ] [rosφ rφ ] [rosφos -φ rosφ φos ] Ug he efnon of rosφ n rφ, we n wre φ P [ ] [ os os ] or he rnsforon r s os [ X ][ ] [ ] os P, 3
Soe Coon Cses of Roon Roon of 9 ouner lokwse ou he orgn os [ ] os Roon of 8 ouner lokwse ou he orgn os os [ ] Roon of 7 ouner lokwse ou he orgn os os In ll he ove ses e[] [ ] Refleon- Spel se of Roon Refleon s spel se of roon of 8 ou lne n plne psg hrough he orgn. Eg ou -s Aou -s [ ] [ ] Aou he lnes n - respevel re: [ ] [ ]
Refleon- Properes If wo pure refleons ou lne psg hrough he orgn re pple suessvel he resul s pure roon. he eernn of pure refleon r s - [ ] Properes of rnsforon Mres De[]? In se of roon, refleon. Wh s he geoerl nerpreon of nverse of []? Show h - [I] os [ ] os os os os os [ ] hus for pure roon e[] he nverse of s s rnspose 5
6 Show h he followng rnsforon r gves pure roon ] [ ] e[ Eple Eple A A un squre s rnsfore rnsforon r. he resulng poson veors re: Deerne he rnsforon r use. Ans: 6 8 3 ] [ 6 3 ] [ ; 6 8 3 ] ][ [ ] [ X
7 Eple Show h he sher rnsforon n n reons ogeher s no he se s sher long followe sher long? Ans: Eple Prole Conser rngle whose veres re, n. Fn he onene rnsforon r n he rnsfore veres for roon of 9 ou he orgn followe refleon hrough he lne -. Coen on he sequene of rnsforons. } }[ [ } }[ }[ [ X X For seeng he effe of hngng he sequene of operons le us reverse he orer,.e, frs refleon n hen roon. } }[ [ } }[ }[ [ X X
8 Eple Prole: Soluon Conu onon of rnsforons n sequene rnsle he rgh-ngle vere o he orgn -, - Roe 5 o π / rn π / os π /.77, 5,,3.8,.8 -.,.,,,. ' ' ' w w ' ' ' os os " " " w w
9 os os os os os os ' ' ' os os " " " w w w w he opuon of [ ] fro [ ] [ ] s lle r ulplon. he generl for s: A sequene of rnsforons n e lupe n gle r v r ulplons h f g e h f g e h g f e Slng relve o fe pon. rnsle he pon-, -n o he orgn. Sle he oje S, S sle fors 3. rnsle he pon k, n, n [ ] [ ] [ ][ ] S [ ] n S S n Colun eors
Roon ou n rrr pon, n [ ] [ ][ ][ ] R [ ] os os n n Row eors Roon ou n rrr pon. rnsle he pon-, -n o he orgn. Roe he oje ou he orgn o 3. rnsle k,n E: Cener of n oje,3, roe ou he ener 9 o CC P [ h] [ ] [ ][ ][ ] R [ ] 7 7 OR P, n [ ] os os n n
Refleon O3 Refleon s spel se of roon SO3 when ngel of roon s 8 ou n s n he plne 8 [ ], os8 n Refleon ou n rrr lne. rnsle, - so h he lne psses hrough he orgn. Roe he lne ou he s - o 3. Refle oje ou he s. Roe k he lne o 5. rnsle k, [] [ ] [ ][ ][ R][ ] [ ] r r / / 5 5 / / 5 5 / / 5 5 / E: Refle rngle,,,6,,6 ou lne 3/5 /5 8/5 /5 3/5 6/5 6 6 / 5 5 /5 8/5 /5 /5 /5 6/5 n Alerne Meho?
M-pon rnsforon Conser srgh lne eween A, n B,. Le us r o rnsfor hs lne ug generl rnsforon r B M E B A F M A [ ] ] [ B A B A [ ] [ ] ' ' M-pon rnsforon Fro n we n onlue h reursvel onnung hs proess, we wll over ll pons on he lne AB n hene her ges on AB orresponng p. Bu generl proof of en pons rnsforon ull rnsfors he whole lne n e gven usgn he seon forul. An pon h ves he lne AB n he ro p:q n e gven s: Neeless o s h he se s pplle o AB. he spel se of pon rnsforon ours when p:q. B M E B A F M A q p q p
rnsforon of nerseng lnes When wo nerseng lnes re rnsfore ug generl rnsforon r he resulng lnes re lso nerseng Conser nerseon lnes he pon of nerseon: rnsfore pon of nerseon: [ X ] [ ] [ X ] E A A B F E F B ' ' [ ] rnsforon of nerseng lnes Oserve hese pons In he ls sle E. A pr of non-perpenulr lnes ge rnsfore o perpenulr lnes A B E B. B nverse - F pr of perpenulra lnes n ge rnsfore o non-perpenulr lnes 3. Suh resul n hve ssrous effe on he resulng geoer F 3
rnsforon of wo prllel lnes When wo prllel lnes re rnsfore ug generl rnsforon r he resulng lnes ren prllel Conser lnes prllel o eh oher eween he en pons A,,B, n CD. Sne he re prllel he slope s he se gven [ ][ ] X Inerseng lnes n rg o rnsforons When o perpenulr lnes rnsfor s perpenulr lnes? Conser he slro n veor prous of wo veors s follows Le us now rnsfor hese veors generl n fn o n ross prous gn We requre for gnue n ngle o ren unhnge os. k k v v os k k r r [ ][ ] [ ][ ] [ ] I OR ; ;
5 rnsforon of Un squre n re Oserve un squre eng sle n shere long n es sulneousl A B D C A D B C P n P ; ] e[ ] e[ s A p A p A, o,,, Are Slng eple A rngle wh veres,,, n -, s rnsfore Are eres fer rnsforon re rnsfore re, sq.uns A 3 [ ] 3 3 3 [X '] sq.uns 8 A A
Geoerl Inerpreon of Overll Slng [ h] [ ] [ s] s s s h< h h> ewng rnsforons I s rnsforon fro worl CS o Sreen CS. he ensons of he WCS e n n hosen sse of uns SI or FPS e he ler s esure n pels.. rnsle he MCS se pon o orgn. Appl he slng neessr o f no he sreen ls n,n n 3. Move he se pon k o s orgnl poson [ ] [ ][ S][ ] [] n n u un n v vn n un vn 6
rnslon Wh re nvrn? Lenghs Angles Whh rnsforon group? Congruen e[] rnslon Dlon Wh re nvrn? Ro of Lenghs Angles Whh rnsforon group? slr e[] 7
rnslon Roon Wh re he nvrns? Lenghs Angles Whh rnsforon group? Congruen e[] Refleon Wh re he nvrns? Lenghs Angles Whh rnsforon group? Congruen e[] - 8
RefleonrnslonGle Wh re he nvrns? Lenghs Angles Whh rnsforon group? Congruen e[] - Gle R R Sher rnsforon Wh re nvrn? prllels - 5 Whh rnsforon group? ffne - 6 9
Projeve rnsforon Wh re nvrn? Inene Cross ros of lenghs Orer of urves 5 - - Isoeres of Squre How n soeres? en Roonl Refleon Cle le of soeres of squre 9 8 7 H D I 9 8 7 H D 9 9 8 7 D D 8 8 7 9 h D D D H D 9 8 7 7 7 9 8 D D H H H D D 8 7 9 h v D D D H D 8 9 7 D D H D 9 7 8 D D D H 7 9 8
rnsforons Possle n fferen Geoeres rnsforon / Geoeres Euln Slr Affne Projeve Roon rnslon Unfor Slng Non-unfor Slng Sher Cenrl Projeon Invrn Qunes n fferen Geoeres rnsforon / Geoeres Euln Slr Affne Projeve Lenghs Angles Ros of Lenghs Prllels Inene Cross-ros of Lenghs