6.2 Improving Our 3-D Graphics Pipeline

Transcription

1 6.2. IMPROVING OUR 3-D GRAPHICS PIPELINE Impovig Ou 3-D Gaphics Pipelie We iish ou basic 3D gaphics pipelie wih he implemeaio o pespecive. beoe we do his, we eview homogeeous coodiaes Homogeeous Coodiaes You will ecall ha i ode o be able o epese aslaios as a mai opeaio, we added a ea coodiae o ou vecos. Tha ea coodiae was always. Thus, a 2D veco (, y) was epeseed as (, y, ). A3D veco (, y, ) was epeseed as (, y,, ). hese wee called homogeeous coodiaes. To see which veco hese vecos i homogeeous coodiaes epeseed, we simply dopped he. The moe geeal deiiio o homogeeous coodiaes does o equie he ea coodiae o be. I ca be ay umbe, hough i usually will o be. This value is used o epese pois a iiiy. The quesio he becomes o kow which veco i Caesia coodiaes a veco i homogeeous coodiaes epeses, whe is added coodiae is o. To see his, we deie he ollowig equivalece elaio. We say ha wo vecos i homogeeous coodiaes ae equivale i ad oly i oe is a muliple o he ohe. Fo eample, (2, 4, 2, 8) is equivale o (, 2,, 4). By equivale, we mea ha hey epese he same veco i he lowe dimesioal space. Thus, he vecos o he om (, y,, W ) o ay value o all epese he same veco i 3D space. Equivalely, he vecos o he om (, y, W ) oayvalueo all epese he same veco i 2D space. I paicula, (,y,,w) ad ( W, y W, W, ) epese he same veco. Bu, we kow ha ( W, y W, W, ) is he epeseaio i homogeeous coodiaes o ( W, y W, ) W i Caesia coodiaes. Thus, we see ha o id he veco i Caesia coodiaes a veco i homogeeous coodiaes epeses, we divide each o is coodiaes by he added coodiae as log as i is o. I ohe wods, (, y,, W ) is equivale o ( ( W, y W, W, ) ad hus epeses W, y W, ) W. The same applies o 2D vecos ad hei epeseaio i homogeeous coodiaes. To make he oaio simple, we will hik o vecos i homogeeous coodiaes as beig o he om (W,Wy,W,W) ad hei epeseaio i Caesia coodiaes will heeoe be (, y,, ) (we divide each coodiae by W ). Whe we divide each coodiae by W,wesayhawe homogeie he veco. To bee udesad wha is happeig, le us look a he geomeic iepeaio o his i he 2D case. A ypical 2D veco is o he om (, y). Is epeseaio i homogeeous coodiaes is (W,Wy,W). Though his epeses a veco i 2D space, i is a 3D objec. I ac, he se o pois {(W,Wy,W) W R} is a lie i 3D space as show o igue 6.8. We homogeie his veco o id is Caesia epeseaio, we ge (, y, ). So, we see ha geomeically, he caesia coodiaes epeseaio o a veco i homogeeous coodiaes is he iesecio o he plae W =ad he lie omed by all he vecos equivale o he give veco i homogeeous

2 82 CHAPTER 6. 3-D GEOMETRY ENGINE Figue 6.8: XY W homogeeous coodiae space coodiaes. I ohe wods, he caesia y plae coespods o he plae W =i homogeeous coodiaes. I we sa wih vecos i he y plae, ad add a homogeeous coodiae o o hem. We he apply some asomaios o hem. I hese asomaios chage he las coodiae (which pespecive will do), geomeically, i meas ha ou veco has bee "lied" om he plae W =. Toakeibackohisplae,iohewodsogeis caesia coodiaes epeseaio, we simply homogeie i (divide each o is coodiaes by he las coodiae) Pespecive Oe o he dawbacks o ohogaphic pojecio is he loss o deph. To solve his, we would like o epese lies which ae uhe apa om he viewe smalle ha lies which ae close. This is called pespecive. Oce agai, hee ae seveal ways o impleme his. The mehod we will use is bes eplaied by lookig a igue 6.9. Le us assume he ollowig:. The gae diecio is he same as he mius ais. 2. The viewe s eye is a e. 3. The view plae is d uis om he viewe s eye. Objecs will be pojeced o he view plae as show i igue 6.9. Usig simila iagles, we ca id he sie o he pojeced objec y s as ollows: y s d = y

3 6.2. IMPROVING OUR 3-D GRAPHICS PIPELINE 83 Theeoe y s = d y (6.9) Similaly, i he diecio, we have s = d (6.) Thus, he sie o a objec will be popoioal o, ha is o he ivese o is disace om he viewe. This is pecisely wha we wee yig o achieve. Figue 6.9: Pespecive Pojecio I he case he gae diecio is o i he diecio o he mius ais, we will is eed o do a chage o coodiaes. Oce we have peomed he pojecio, we sill eed o map eveyhig oo he scee. I ohe wods, besides pespecive, we eed o also peom a lo o he asomaios we have aleady implemeed. Theeoe, as we impleme pespecive, we would like o do i i a way ha especs he ollowig:. We would like o use he machiey we have aleady developed, wheeve i is eeded. Pespecive would jus be a added sep o wha we have aleady doe. 2. Because wha we have aleady doe has bee implemeed as mai asomaios, pespecive should also be implemeed as a mai asomaio, ohewise i will o be compaible wih he ohe asomaios. Pespecive as a added sep o ou machiey The idea is o is pojec eveyhig o he view plae, as show o igue 6.9. The, we will apply ou machiey o he objecs o he view plae. Because

4 84 CHAPTER 6. 3-D GEOMETRY ENGINE pespecive is a pojecio alog a lie hough he viewe s eye, all he pois o his lie will be pojeced o he same poi o he view plae. Ohogaphic pojecio is a pojecio paallel o he gae diecio. Thus, pois o a lie paallel o he gae diecio will be mapped o he same poi. Thus, i we camappoisoaliehoughheviewe seyeopoisoaliepaallel o he gae diecio, we ca he apply ou ohogaphic pojecio o iish he wok. This is illusaed o igue 6..Because we will eveually dop Figue 6.: Pespecive pojecio as a ohogaphic pojecio he coodiae, i does o mae how he coodiaes o he pois ae asomed. Though, we would like o peseve ode. Ou goal is he o do he ollowig: Coside a lie L hough he viewe s eye. Le P be he poi a which i iesec he view plae. Ceae a mai asomaio which will map all he pois o L o pois o a lie hough P paallel o he gae diecio. We ca he apply ou peviously developed machiey o iish he wok. Beoe we poceed uhe, le us ioduce some emiology. Deiiio 84 The plae = will be called he view plae o he ea plae. Deiiio 85 The plae = will be called he a plae. Deiiio 86 The poio o he pyamid bewee he plaes = ad = o he op image o igue 6. is called he view usum.

5 6.2. IMPROVING OUR 3-D GRAPHICS PIPELINE 85 Figue 6.: Wold hough he viewe s eye mapped oo he ohogaphic cube

6 86 CHAPTER 6. 3-D GEOMETRY ENGINE Figue 6.2: The pespecive pojecio maps lie hough he viewe s eye o lies paallel o he ais. Figue 6.3: Oigial Cube ad he View Fusum

7 6.2. IMPROVING OUR 3-D GRAPHICS PIPELINE 87 The pespecive pojecio will leave pois o he ea plae uchaged. Figues 6., 6.2 ad 6.3 illusae his cocep. The pois iesecig he a plae ad he view usum will be mapped o pois iesecig he a plae ad he oigial cube. I paicula, he ollowig coe should emai uchaged: l b, l, b ad l l o, will be mapped o we ow develop he pespecive asomaio..also, will be mapped ad so o. Wih his i mid, Pespecive as a mai asomaio The quesio le o solve is how o map pois o L o pois o a lie hough P paallel o he gae diecio, so ha he mappig is a mai asomaio. A is glace, his appeas o be diicul because, as equaio 6.9 shows, his asomaio will ivolve dividig by. Bu cao be pa o he mai o he asomaio. This mai has o be idepede o he pois we ae asomig. This is whee ou homogeeous coodiaes will come io play. Le M p deoe he pespecive mai we eed o id. I is a 4 4 mai. Thus, we eed o id is 6 eies. This meas solvig a sysem o 6 equaios i 6 ukows. I ealiy, i is o so diicul. Fom equaios 6.9 ad 6., ad usig he ac ha he view plae is = (meaig d i hose equaios will be ), we see ha a poi o coodiaes y should be mapped o y?. We pu a quesio mak o he coodiaes, because i does o eally mae sice i will be dopped ayway. The poblem wih his asomaio is ha appeas as a aco o boh ad y, which meas ha i should be pa o he mai M p, which is o possible. Tha would make M p depede o each poi beig asomed. A aleaive is i we coside he poi y? as beig he esul o havig bee homogeied (divide is coodiaes by w). Which we would have o do i he M p mai had modiied he w coodiae o he iiial poi. Fo eample, i applyig M p o y had poduced y?, he homogeiig i would poduced he desied esul, wihou havig o be

8 88 CHAPTER 6. 3-D GEOMETRY ENGINE i he mai. So, we ae seekig a mai M p such ha M p The ollowig mai would wok M p = α β y = We oly eed o id wo eies, α ad β. To id hem, we use equaios, we use he ac ha emais uchaged by he asomaio while, will be mapped o. We is eed o compue simila iagles, as show i igue 6.9, we see ha y?.. Usig = = So, his ells us ha ou asomaio should do he ollowig M p = ad M p = i ohe wods M p =

9 6.2. IMPROVING OUR 3-D GRAPHICS PIPELINE 89 Theeoe, we have: M p = = α β α + β Also, M p = = I we homogeie his poi, we ge which is he same as α β α + β (α + β) (α + β) So,wegeheollowigwoequaios { α + β = (α + β) = Thesoluioohissysemisα = ( + ),β =. Thus, we see ha he pespecive mai is: M p = + We lis some o he popeies o his pespecive mai.

10 9 CHAPTER 6. 3-D GEOMETRY ENGINE Poposiio 87 The pespecive mai has he popeies we said we would like i o have. I paicula, i escales he ad y coodiaes o pois by a aco o ad i leaves pois o he plae = uchaged. Poo. The eec o he mai o a veco (, y, ) is y + = y + ( + ) I we homogeie he poi, we obai y + We see ha ad y have bee escaled by a aco o. Wealsoseehaa poi o he plae = will o be aeced by his asom. Such a poi would have coodiaes (, y, ). We have + y = y Remak 88 Whe we homogeie a poi o a veco, we divide each o is coodiaes by he same quaiy. As log as he pespecive mai we apply poduces equivale vecos i homogeeous coodiaes, he pocess will wok. Whe we homogeie hese vecos, i hey ae equivale hey will esul i he same homogeied veco. Theeoe, i M is a pespecive mai, ad h is a cosa, he hm will also wok as a pespecive mai. Le us assume ha M asoms a veco v io aohe veco w. Thais The, Mv = w (hm) v = h (Mv) = hw Bu w ad hw ae equivale ad hus esul i he same homogeied poi. This suggess ha we ca muliply ou mai M p by a cosa. I we muliply M p by, i will make i ice. Thus, o M p we will acually use M p = +

11 6.2. IMPROVING OUR 3-D GRAPHICS PIPELINE 9 Someimes, i may be ecessay o evese he eec o pojecio. Fo his, we would use Mp. Wih his simpliied om o M p, he ivese is also easie o id. The eade will check ha Mp = + Fo he same easos as above, we ca also muliply M p cosa. A simple vesio o Mp Thus, o Mp,wewilluse Puig i ogehe M p = by ay o eo by. is obaied by muliplyig M p + Oce he pespecive mai has bee applied, we ca he apply he ohogaphic asomaio developed ealie. he mai ecodig his was called M. I he view posiio ad he gae diecio ae o sadad, we may eed o apply a coodiae asom is. The mai ecodig his was called M v. The asomaio which will peom all he seps descibed is M M p M v. We have he ollowig algoihm: Algoihm 89 To daw 3D lies wih ed pois a i ad b i usig pespecive, do he ollowig:. Compue M 2. Compue M v 3. Compue M p 4. Se M = M M p M v 5. Fo each i, do p = Ma i q = Mb i dawlie ( p W p, y p W p, q W q, y q W q ) I he above algoihm, W q deoes he ouh coodiae o ou poi ae i has bee asomed.

12 92 CHAPTER 6. 3-D GEOMETRY ENGINE Deiiio 9 The mai poduc M o M p is oe called he pojecio mai. We have M pojecio = 2 l+ l 2 b Popeies o he pespecive asom l b+ b + 2 The pespecive asom maps lies o lies. The pespecive asom maps plaes o plaes. The pespecive asom maps lie segmes o lie segmes, pesevig he odeig o he pois. Theeoe, he pespecive asom akes veices ad edges o a iagle io veices ad edges o aohe iagle Field-o-View Wih he machiey we have developed, he use mus speciy he ollowig quaiies:. The viewe s eye posiio 2. The gae diecio ad he up veco 3. The cube i which ou wold leaves. This ca be doe by speciyig (l,,, b,, ). I is ulikely we will be able o simpliy wha we have o speciy i ad 2. Le us see wha we ca do abou 3. I we had a simple sysem which looked hough he cee o he widow, he we would have l = b = This aleady deceases by 2 he umbe o quaiies we eed o speciy. I i addiio we equie piels o be squae, i ohe wods = y The, his deceases by he umbe o quaiies we eed. Wih hese esicios, we eed o be give (,, ). is oe se by speciyig he ield-o-view, show as θ i igue 6.4. Usig simple igoomey, we see ha a θ 2 = Theeoe, i, ad θ ae give, we ca deive all he ohe quaiies usig he assumpios (see poblems).

13 6.3. ASSIGNMENT 93 Deiiio 9 (Field-o-view) The ield-o-view θ is he agle om he boom o he scee o he op o he scee, daw om he viewe s eye posiio. Figue 6.4: The ield o view θ is he agle om he boom o he scee o he op o he scee measued om he eye 6.3 Assigme. Veiy ha Mp give above is coec. 2. Wie dow a pespecive mai o =ad =2. 3. Veiy ha M pojecio akes (,, ) o (,, ). 4. Wie M, M p ad M pojecio i ems o, ad θ. whe he ield-oview is speciied. Use he assumpios o his secio.