Projection of geometric models

Size: px

Start display at page:

Download "Projection of geometric models"

Corey Randall
5 years ago
Views:

1 ojecion of geomeic moels YZU Opimal Design Laboao. All ighs eseve. Las upae: Yeh-Liang Hsu (-9-). Noe: his is he couse maeial fo ME55 Geomeic moeling an compue gaphics, Yuan Ze Univesi. a of his maeial is aape fom CAD/CAM heo an acice, b Ibahim Zei, McGaw-Hill, 99. his 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. anslaion, scaling, mioing, an oaion of a poin ae epesene b he following equaions especivel: () S M () (3) R Z (4) While he las hee equaions ae in he fom of mai muliplicaion, anslaion akes he fom of veco aiion. his 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 homogeneous cooinaes povies an effecive wa o unif he escipion of geomeic ansfomaions as mai muliplicaions. hp://esigne.mech.u.eu.w/

2 ojecion of geomeic moels hp://esigne.mech.u.eu.w/ In hee-imensional space, a poin wih Caesian cooinaes (,, ) has he homogeneous cooinaes (,,, h), whee h is an scala faco. he wo pes of cooinaes ae elae o each ohe b he following equaions: h h h (5) Fo eample, (,, 3, ) an (, 4, 6, ) ae he same poin. anslaion can hen be epesene b mai muliplicaion using homogeneous cooinaes. (6) D (7) An he ansfomaion maices in Equaion (), (3), (4) become s s s S (8) M (9) R () Epessing posiions in homogeneous cooinaes allows us o epesen all geomeic ansfomaion equaions as mai muliplicaions. Fo eample,

3 ojecion of geomeic moels hp://esigne.mech.u.eu.w/ 3 D R D Z () whee D () cos sin sin cos Z R (3) D (4) cos sin cos sin sin cos sin cos R D D (5) o in a geneal fom, (6) he 3 3 submai [ ] pouces scaling, eflecion, o oaion. he 3 column mai [ ] geneaes anslaion. he 3 ow mai [ 3 ] pouces pespecive pojecion, which will be iscusse lae.

4 ojecion of geomeic moels Assignmen Ceae a ecangle in - plane in ou CAD sofwae; 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. 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 Use wo-imensional oaion maices o show ha oaion is no commuaive, ha is, R R R R. Homogeneous cooinaes was inouce b Augus Feinan Möbius in 87. he have he avanage ha he cooinaes of a poin, even hose a infini, can be epesene using finie cooinaes. Ofen fomulas involving homogeneous cooinaes ae simple an moe smmeic han hei Caesian counepas. Homogeneous cooinaes have a ange of applicaions, incluing compue gaphics an 3D compue vision, whee he allow affine ansfomaions an, in geneal, pojecive ansfomaions o be easil epesene b a mai.. ojecions of geomeic moels. 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. o efine a pojecion, a cene of pojecion an a pojecion plane mus be efine. hee 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 4 hp://esigne.mech.u.eu.w/

5 ojecion of geomeic moels 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. ojecos + Cene of pojecion ojecos Cene of pojecion a infini (a) especive pojecion (b) aallel pojecion Figue. pes 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? hee 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. hee ae wo pes of ohogaphic pojecions. he 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. he muli-view 5 hp://esigne.mech.u.eu.w/

6 ojecion of geomeic moels pojecion -- fon, op, an igh views ha ae use cusomail in engineeing awings belong o his pe. he 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. his pe is calle aonomeic pojecions. he peseve paallelism of lines bu no angles. Aonomeic pojecions ae fuhe ivie ino imeic, imeic an isomeic pojecions. he isomeic pojecion is he mos common aonomeic pojecion. he isomeic pojecion has he useful pope ha all hee pincipal aes ae equall foeshoene. heefoe measuemens along he aes can be mae wih he same scalehus 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 ( each) wih one anohe when pojece ono he pojecion plane. he oblique pojecion places he pincipal face of he objec paallel o he plane of he pape, an is ofen use in feehan skeching. he avanage is ha eails on he fon face of he objec eain hei ue shape. he 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.. 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. his 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) he 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. 6 hp://esigne.mech.u.eu.w/

7 ojecion of geomeic moels R (8) whee [R] an ae he oaional an anslaional mapping pas of [] especivel. heefoe, Equaion (8) gives he posiion veco of he oigin of he ssem as an is oienaion as [R] boh measue in he ssem. he 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. he Z v ais efines he viewing iecion. he 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. View efiniion. o 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 7 hp://esigne.mech.u.eu.w/

8 ojecion of geomeic moels ono ha plane. hus, a view of a moel is geneae in wo seps: oae he moel popel an hen pojec i..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 oaion. o obain he fon view, we onl (no oaion is neee) nee o se = fo all he ke poins of he moel. hus, Equaion (8) becomes An (9) v () whee v is he poin epesse in he VCS. Fo he fon view, Equaion () gives v = an v =. 8 hp://esigne.mech.u.eu.w/

9 ojecion of geomeic moels Fon op Righ X Y v Z (a) Moel views elaive o is MCS Y Y v op X v,x Fon op 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 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. he cooinae is he one o se o eo because he Y ais of he MCS coincies wih he pojecion iecion. In his case, [] becomes () An Equaion () 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 =-. 9 hp://esigne.mech.u.eu.w/

10 ojecion of geomeic moels he 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. hus, () which gives v =- an v =. Eamining Equaion (), (), an () shows ha [] is a singula mai wih a column of eos which coespons o he MCS ais ha coincies wih he pojecion o viewing iecion. o obain he isomeic pojecion o view, he moel an is MCS ae cusomail oae an angle 45 abou he Y v ais followe b a oaion abou he X v ais. In pacice, he angle is aken as iangles in manual consucion of isomeic views. 3 o enable he afing (plasic) cos sin cos sin v (3) sin cos sin cos 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 (3) o ansfom he 8 ke poins of he block. 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..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. he 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. hp://esigne.mech.u.eu.w/

11 ojecion of geomeic moels hp://esigne.mech.u.eu.w/ he ansfomaion mai of cooinaes of poins fom he VCS o he ECS o vice vesa can be wien as (4) his 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 / (5) v / (6) / v (7) If his equaion is epane i gives. / v

12 ojecion of geomeic moels Figue 4. especive pojecion along he Z v ais Assignmen 6 Use he same 3D block in Assignmen 5. Wie a pogam in Malab o appl Equaion (7) 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/

Projection of geometric models

Projection of geometric models 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