ojecion of geomeic moels Eie: Yeh-Liang Hsu (998-9-2); ecommene: Yeh-Liang Hsu (2-9-26); las upae: Yeh-Liang Hsu (29--3). Noe: This is he couse maeial fo ME55 Geomeic moeling an compue gaphics, Yuan Ze Univesi. a of his maeial is aape fom CAD/CAM Theo an acice, b Ibahim Zei, McGaw-Hill, 99. This maeial is be use sicl fo eaching an leaning of his couse. ojecion of geomeic moels. Homogeneous cooinae Man gaphics applicaions involve sequences of geomeic ansfomaions. An animaion, fo eample, migh equie an objec o be anslae an oae a each incemen of he moion. Hee we consie how he mai epesenaions can be efomulae so ha such ansfomaion sequences can be efficienl pocesse. Tanslaion, scaling, mioing, an oaion of a poin ae epesene b he following equaions especivel: + [ S] [ M] () (2) (3) [ R ] Z (4) While he las hee equaions ae in he fom of mai muliplicaion, anslaion akes he fom of veco aiion. This makes i inconvenien o concaenae ansfomaions involving anslaion. I is esiable, heefoe, o epess all geomeic ansfomaions in he fom of mai muliplicaions onl. Repesening poins b hei hp://esigne.mech.u.eu.w/
ojecion of geomeic moels homogeneous cooinaes povies an effecive wa o unif he escipion of geomeic ansfomaions as mai muliplicaions. In hee-imensional space, a poin wih Caesian cooinaes (,, ) has he homogeneous cooinaes (,,, h), whee h is an scala faco. The wo pes of cooinaes ae elae o each ohe b he following equaions: h h h (5) Fo eample, (, 2, 3, ) an (2, 4, 6, 2) ae he same poin. Tanslaion can hen be epesene b mai muliplicaion using homogeneous cooinaes. (6) T (7) D An he ansfomaion maices in Equaion (2), (3), (4) become (8) s s s S (9) ± ± ± M () 33 32 3 23 22 2 3 2 R Epessing posiions in homogeneous cooinaes allows us o epesen all geomeic ansfomaion equaions as mai muliplicaions. Fo eample, hp://esigne.mech.u.eu.w/ 2
[ D ][ R ][ D ] [ T] ojecion of geomeic moels 2 Z () whee [ D ] (2) R Z cosθ sinθ sinθ cosθ (3) [ D ] 2 (4) o in a geneal fom, [ D ][ R ][ D ] ( cosθ ) sin sinθ + ( cosθ ) cosθ sinθ θ sinθ cosθ T 2 (5) T T 2 22 23 24 2 T 3 4 2 32 42 3 33 43 4 34 44 3 T (6) The 3 3 submai [T ] pouces scaling, eflecion, o oaion. The 3 column mai [T 2 ] geneaes anslaion. The 3 ow mai [T 3 ] pouces pespecive pojecion, which will be iscusse lae. Assignmen Ceae a ecangle in - plane; specif he cooinaes of he 4 cone poins. Assume some paamees an appl he following wo-imensional ansfomaions, anslaion, oaion, anslaion, o he objec in ou CAD sofwae. Daw he objec afe hese ansfomaions. 3 hp://esigne.mech.u.eu.w/
ojecion of geomeic moels Use homogenous cooinaes o epesen he ke poins of he objec ou geneae. Wie a pogam in Malab an concaenae he 3 ansfomaions ino one ansfomaion mai as shown in Equaion ()~(5). Calculae he posiions of he ke poins using his ansfomaion mai. Regeneae he objec b connecing he ke poins afe ansfomaion. Does he objec mach he objec ou ew in ou CAD sofwae? Assignmen 2 Use wo-imensional oaion maices o pove ha oaion is no commuaive, ha is, [ R ] [ R ][ R ] R 2 2. Homogeneous cooinaes have been use in compue gaphics an geomeic moeling fo a long ime. The ae useful o obain pespecive views of geomeic moels. The subjecs of pojecive geome, mechanism analsis an esign, an oboics uilie homogeneous cooinaes quie ofen in evelopmen an fomulaion. The em homogenous cooinaes is use in mahemaics o efe o he effec of his epesenaion on Caesian equaions. When a Caesian poin (,, ) is convee o a homogenous epesenaion (,,, h), equaions conaining,, an, such as f(,, ), become homogenous equaions in he fou paamees,,, an h. This jus means ha if each of he fou paamees is eplace b an value v imes ha paamee, he value v can be facoe ou of he equaions. 2. ojecions of geomeic moels 2. especive pojecion an paallel pojecion Viewing a hee-imensional moel is a ahe comple pocess ue o he fac ha ispla evices can onl ispla gaphics on wo-imensional sceens. To efine a pojecion, a cene of pojecion an a pojecion plane mus be efine. Thee ae wo iffeen pes of pojecions base on he locaion of he cene of pojecion (o pojecion efeence poin) elaive o he pojecion plane, as shown in Figue. If he cene is a a finie isance fom he plane, pespecive pojecion esuls an all he pojecos mee a he cene. If, on he ohe han, he cene is a an infinie isance, all he pojecos become paallel (mee a infini) an paallel pojecion esuls. 4 hp://esigne.mech.u.eu.w/
ojecion of geomeic moels ojecos + Cene of pojecion 2 2 ojecos 2 2 Cene of pojecion a infini (a) especive pojecion (b) aallel pojecion Figue. Tpes of pojecions especive pojecion oes no peseve paallelism, ha is, no wo lines ae paallel. especive pojecion ceaes an aisic effec ha as some ealism o pespecive views. Sie of an eni is invesel popoional o is isance fom he cene of pojecion; ha is, he close he eni o he cene, he lage is sie is. aallel pojecion peseves acual imensions an shapes of objecs. I also peseves paallelism. Angles ae peseve onl on faces of he objec ha ae paallel o he pojecion plane. Assignmen 3 Does ou CAD sofwae suppo pespecive pojecion? If so, buil a 3D objec o emonsae he iffeence beween pespecive pojecion an paallel pojecion. In ou CAD sofwae, how o ou moif he paamees o change he viewpoin of he pespecive pojecion? Thee ae wo pes of paallel pojecions base on he elaion beween he iecion of pojecion an he pojecion plane. If his iecion is nomal o he pojecion plane, ohogaphic pojecion an views esul. If he iecion is no nomal o he plane, oblique pojecion occus. Thee ae wo pes of ohogaphic pojecions. The mos common pe is he one ha uses pojecion planes ha ae pepenicula o he pincipal aes of he MCS of he moel; ha is, he iecion of pojecion coincies wih one of hese aes. The muliview pojecion -- fon, op, an igh views ha ae use cusomail in engineeing awings belong o his pe. The ohe pe of ohogaphic pojecion uses pojecion planes ha ae no nomal o a pincipal ais an heefoe show seveal faces of a moel a once. This pe is calle aonomeic pojecions. The peseve paallelism of lines bu no angles. 5 hp://esigne.mech.u.eu.w/
ojecion of geomeic moels Aonomeic pojecions ae fuhe ivie ino imeic, imeic an isomeic pojecions. The isomeic pojecion is he mos common aonomeic pojecion. The isomeic pojecion has he useful pope ha all hee pincipal aes ae equall foeshoene. Theefoe measuemens along he aes can be mae wih he same scale -hus he name: iso fo equal, meic fo measue. In aiion, he nomal o he pojecion plane makes equal angles wih each pincipal ais an he pincipal aes make equal angles (2 each) wih one anohe when pojece ono he pojecion plane. The oblique pojecion places he pincipal face of he objec paallel o he plane of he pape, an is ofen use in feehan skeching. The avanage is ha eails on he fon face of he objec eain hei ue shape. The isavanage is ha oblique pojecion oes no appea ealisic. Assignmen 4 Daw a famil ee fo he pes of pojecions iscusse above. Use ou CAD sofwae o geneae all pojecions in he famil ee using he 3D objec in Assignmen 3. 2.2 Mappings of geomeic moels o he viewing cooinae ssem Mapping of a poin (o a se of poins) belonging o an objec fom one cooinae ssem o anohe is efine as changing he escipion of he poin (o he se of poins) fom he fis cooinae ssem o he secon one. This is equivalen o ansfoming one cooinae ssem o anohe. Given he cooinaes of a poin measue in a given cooinae ssem, fin he cooinaes of he poin measue in anohe cooinae ssem, sa such ha f(, ansfomaion paamees) (7) The mapping paamees escibe he elaionship beween he wo ssems an consis of he posiion of he oigin an oienaion of he ssem elaive o he ssem. [ T] 2 3 2 22 32 3 23 33 [ R] (8) whee [R] an ae he oaional an anslaional mapping pas of [T] especivel. 6 hp://esigne.mech.u.eu.w/
ojecion of geomeic moels Theefoe, Equaion (8) gives he posiion veco of he oigin of he ssem as an is oienaion as [R] boh measue in he ssem. The columns of [R] ae he componens of he uni vecos of he ssem (along is aes) measue in he ssem. A view has a viewing cooinae ssem (VCS). I is a hee-imensional ssem wih he X v ais hoional poining o he igh an he Y v ais veical poining upwa, as shown in Figue 2. The Z v ais efines he viewing iecion. The posiive Z v ais has an opposie sense o he viewing iecion o keep he VCS a igh-hane cooinae ssem, even hough a lef-hane ssem ma be moe esiable hee since is posiive Z v ais is in he iecion of he lines of sigh emiing fom he viewing ee. Viewing plane Y v Viewpo (view winow) View oigin X v Z v Viewing iecion Viewing ee o Figue 2. View efiniion. To obain views of a moel, he viewing plane, he X v Y v plane, is mae coincien wih he - plane of he MCS such ha he VCS oigin is he same as ha of he MCS. Moel views now become a mae of oaing he moel wih espec o he VCS aes unil he esie moel plane coincies wih he viewing plane followe b pojecing he moel ono ha plane. Thus, a view of a moel is geneae in wo seps: oae he moel popel an hen pojec i. 2.3 Ohogaphic pojecions Figue 3 shows he elaionship beween MCS an VCS. An ohogaphic pojecion (view) of a moel is obaine b seing o eo he cooinae value coesponing o he MCS ais ha coincies wih he iecion of pojecion (o viewing) afe he moel 7 hp://esigne.mech.u.eu.w/
ojecion of geomeic moels oaion. To obain he fon view, we onl (no oaion is neee) nee o se fo all he ke poins of he moel. Thus, Equaion (8) becomes [ T] (9) An v [ T] (2) whee v is he poin epesse in he VCS. Fo he fon view, Equaion (2) gives v an v. Fon Top Righ X Y v Z (a) Moel views elaive o is MCS Y Y v Top X v,x Fon Top Righ X v Z Z Z v X Y v,y Y v,y Fon Righ Fon view X v,x Z Righ view X v (b) MCS an VCS elaionship Figue 3. Relaionship beween MCS an VCS 8 hp://esigne.mech.u.eu.w/
ojecion of geomeic moels Fo he op view, he moel an is MCS ae oae b 9 abou he X v ais followe b seing he cooinae of he esuling poins o eo. The cooinae is he one o se o eo because he Y ais of he MCS coincies wih he pojecion iecion. In his case, [T] becomes [ T] (2) An Equaion (2) gives v an v -. If we use he above equaion o ansfom he MCS iself, he X ais () ansfoms o v an he Y ais () ansfoms o v -. The igh view shown in Figue 3 can be obaine b oaing he moel an is MCS abou he Y v ais b -9 an seing he cooinae o eo. Thus, [ T] (22) which gives v - an v. Eamining Equaion (2), (2), an (22) shows ha [T] is a singula mai wih a column of eos which coespons o he MCS ais ha coincies wih he pojecion o viewing iecion. To obain he isomeic pojecion o view, he moel an is MCS ae cusomail oae an angle θ ± 45 abou he Y v ais followe b a oaion φ ± 35. 26 abou he X v ais. In pacice, he angle φ is aken as ± 3 o enable he afing (plasic) iangles in manual consucion of isomeic views. cosθ sinθ cosφ sinφ v T T (23) sinφ cosφ sinθ cosθ Assignmen 5 Ceae a 3D block an assign cooinaes o all 8 cone poins of he block Wie a pogam in Malab an use Equaion (23) o ansfom he 8 ke poins of he block. 9 hp://esigne.mech.u.eu.w/
ojecion of geomeic moels Seing cooinaes of he ke poins o eo, hen aw he block again. Do ou obain an isomeic view of he block? Show ou Malab pogam oo. 2.4 especive pojecions One common wa o obain a pespecive view is o place he cene of pojecion along he Z v ais of he VCS an pojec ono he v o he X v Y v plane. A new cooinae ssem calle he ee cooinae ssem (ECS) is inouce elaive o he line of sigh. The ECS has an oigin locae a he same posiion as he viewing ee. Is X e an Y e aes ae paallel o he X v an Y v aes of he VCS. Howeve, i is a lef-hane ssem. The ansfomaion mai of cooinaes of poins fom he VCS o he ECS o vice vesa can be wien as T (24) This mai simpl inves he sign of he cooinae. In he ohogaphic views, he ECS is locae a infini. I is obvious ha he ECS can be eplace b he VCS. In his case, poins wih smalle values ae inepee as being fuhe fom he viewing ee. One-poin pojecion is shown in Figue 4. Fom simila iangles, we can he following equaions. v (25) / v 2 (26) / v / (27) If his equaion is epane i gives v T [ ( / )]. hp://esigne.mech.u.eu.w/
ojecion of geomeic moels Figue 4. especive pojecion along he Z v ais Assignmen 6 Use he same 3D block in Assignmen 5. VZ Wie a pogam in Malab o appl Equaion (27) o is 8 ke poins, hen ceae he pespecive pojecion of he block. Change seveal values of, how oes he pespecive pojecion change? Show ou Malab pogam oo. hp://esigne.mech.u.eu.w/