3D Viewing and Projec5on. Taking Pictures with a Real Camera. Steps: Graphics does the same thing for rendering an image for 3D geometric objects

Transcription

1 3D Vieing and Projec5on Taking Pictures ith a Real Camera Steps: Iden5 interes5ng objects Rotate and translate the camera to desired viepoint Adjust camera seings such as ocal length Choose desired resolu5on and aspect ra5o, etc. Take a snapshot Graphics does the same thing or rendering an image or 3D geometric objects

2 3D Geometr Pipeline Beore being turned into piels b graphics hardare, a piece o geometr goes through a number o transorma5ons... Model space (Object space) 3D Geometr Pipeline Beore being turned into piels b graphics hardare, a piece o geometr goes through a number o transorma5ons... World space 2

3 3D Geometr Pipeline Beore being turned into piels b graphics hardare, a piece o geometr goes through a number o transorma5ons... Ee space (Vie space) 3D Geometr Pipeline Beore being turned into piels b graphics hardare, a piece o geometr goes through a number o transorma5ons... Normalied projec5on space 3

4 3D Geometr Pipeline Beore being turned into piels b graphics hardare, a piece o geometr goes through a number o transorma5ons... Image space, indo space, raster space, screen space, device space 3D Geometr Pipeline Object space World space Vie space Image space Normalied projec5on space 4

5 5 3D Geometr Pipeline World space Object space Translate, scale, rotate gltranslate*(t,t,t ) = ʹ ʹ ʹ t t t = ʹ ʹ ʹ s s s glscale*(s,s,s ) 3D Geometr Pipeline World space Object space Translate, scale, rotate glrotate* ),,, ( v v v θ ), (, v v v r = T R R R R R T ) ( ) ( ) ( ) ( ) ( α β θ β α Rotate about r b angle θ R (α) = cos α sin α -sin α cos α

6 3D Geometr Pipeline Object space World space Vie space Image space Normalied projec5on space 3D Geometr Pipeline Net e look at ho e ould compute the orld to ee (vie) transorma5on Translate, rotate World space Vie space 6

7 The Vieing Sstem Three aspects o the vieing process:. Posi5oning the camera SeIng the MODELVIEW matri 2. Selec5ng a lens SeIng the PROJECTION matri 3. Clipping SeIng the vie volume Start ith a discussion o OpenGL deaults. OpenGL Vieing Deaults 7

8 The OpenGL Camera In OpenGL, ini5all the orld and camera rames are the same The camera is located at origin and points in the - direc5on OpenGL also speciies a deault vie volume that is a cube ith sides o length 2 centered at the origin: glortho(-,, -,, -,); The OpenGL (deault) Projec5on Deault projec5on is orthogonal (orthographic) Deault projec5on matri is iden5t clipped out ( coordinates dropped) 8

9 Orthographic Projec5on in OpenGL glortho(min, ma, min, ma, min, ma); We alas vie in - direc5on min and ma are speciied as posi&ve distances along, rela&ve to the camera Hands- on Session The OpenGL tutor programs Go to the class ebsite, click on the Links sec5on Donload and compile the OpenGL tutors Run the projec&on tutor Use the menu to select orthographic projec5on Pla ith the parameters o glortho. Double the vieing volume 2. Halve the vieing volume 9

10 Posi5oning the Camera OpenGL vieing deaults have limita5ons: ied origin and ied projec5on direc5on Ho to obtain arbitrar camera orienta5ons and posi5ons? Posi5oning the Camera Suppose that e ish to posi5on the camera at (,, 2).r.t. the orld. To (equivalent) possibili5es: Transorm the orld prior to crea5on o objects: gltranslate(,, - 2); Posi5on the camera ith respect to the orld: glulookat(,, 2, );

11 Posi5oning the Camera OpenGL Code Moving the objects: glmatrimode(gl_modelview) glloadidentit(); gltranslate(.,.,-2.); Moving the camera: glmatrimode(gl_modelview); glloadidentit(); glulookat(.,., 2.,.,.,.,.,...); Posi5oning the Camera Vie Parameters Camera loca&on (ee): posi5on in the orld coordinates Vieing direc&on (at): hich direc5on are aim the camera Camera orienta&on (up) up up up (at, at, at) glulookat(ee, ee, ee, at, at, at, up, up, up);

12 Camera Coordinate Sstem up vector (up,up,up ) vie direc5on N at ee back vector (back, back, back ) right vector (right, right, right ) glulookat(ee, ee, ee, at, at, at, up, up, up ); Ho to determine back and right? Camera Coordinate Sstem glulookat(ee, ee, ee, at, at, at, up, up, up ); up vector (up,up,up ) vie direc5on N at ee back vector (back, back, back ) right vector (right, right, right ) N = at ee N N back = up back right = up 2

13 Camera Coordinate Sstem Mapping rom orld to ee coordinates ee posi5on maps to origin right vector maps to the ais up vector maps to the ais back vector maps to the ais up ee back right Vieing Transorma5on We have the camera in orld coordinates We ant to transorma5on T hich takes object rom orld to camera c p = T, c Trick: ind T - taking object rom camera to orld p p = (T,c ) p c 3

14 Vieing Transorma5on Trick: take points rom camera to orld sstem align right ith, up ith, back ith up ee back right Vieing Transorma5on Trick: take points rom camera to orld sstem align right ith, up ith, back ith up back right 4

15 Vieing Transorma5on Trick: ind T - taking object rom camera to orld ee posi5on maps to origin back vector maps to ais up vector maps to ais right vector maps to ais back up right " right up back ee right = T c, = up back ee right up back ee # T,c % & Understanding glulookat 2- unit cube centered at (,,) glulookat(,,,,,,,, ) Change the ee posi5on to get these: glulookat(,,,,,,,,) 5

16 glulookat Up- vector Eects Sie 2 cube centered at the origin, vieed b: glmatrimode(gl_projection); glloadidentit(); glortho(-4.,4., -4., 4., -4., 4.); glmatrimode(gl_modelview); glloadidentit(); glulookat(,,,,,,,, ); Vieed b: glulookat(,,,,,,,, ); Hands- on Session Run again the projec&on tutor Use the menu to select orthographic projec5on Double the vieing volume Pla ith the parameters o glulookat. Modi the ee posi5on 2. Modi the LookAt point (center) posi5on 3. Modi the up vector. (Turn the image upside- don.) 6

17 Perspec5ve Projec5on We have determined ho objects are placed rela5ve to camera. But ho are the objects projected to the image? (Conver5ng rom 3D to 2D) Ho Do We See the World? Images are inverted on the re5na 7

18 Relected Light The colours that e perceive are determined b the nature o the light relected rom an object For eample, i hite light is shone onto a green object most avelengths are absorbed, hile green light is relected rom the object White Light Green Light Colours Absorbed Pinhole Camera Model Pinhole camera - bo ith a 5n ront hole and ilm at the back Image is upside don - - models hat our ees do 8

19 Pinhole Camera Principle (,, ) I = (?,?,?) lens (ocal length) Determine point I = (?,?,?) here projec5on ra intersects the image. Mathema5cs o Perspec5ve Determine point I = (?,?,?) here the projec5on ra intersects the image. One coordinate is eas to determine: I = (?,?, ) (,, ) I = (?,?,?) lens 9

20 2 Consider the line passing through origin and parallel to vector v. Line through origin parallel to v is the set o all points ith: Aside Mathema5cs Warmup: Parametric Equa5ons or Lines = 2 v = Aside Eercise: Give parametric equa5on or line parallel to passing through : 5

21 2 Aside Parametric Equa5ons in 3D = v = 7 2 A = B Line through A in the direc5on o B is: Line through (,,) parallel to (,,) is: Back to Camera Determine point I = (?,?,) here the projec5on ra intersects the image. (,, ) I = (?,?, ) lens

22 Projec5on Ra Parametric equa5on or projec5on ra or orld point (,, ): = t B varing t, e can travel along the line. What value o t puts us on the image (makes = )? Projec5on Ra No subs5tute or and : = t = " I = # = t =,, ) 22

23 23 Virtual Image in Front o Camera To simpli things, e orm the image in ront o the pinhole: (,, ) lens I =,,! " # # % & & Image point = (?,?,?) Ho are and coordinates aected b values o? Projec5on Matri =???????????????? ( ),, ),, maps to What is the 44 transorma5on matri? M perspec;ve

24 24 Projec5on Matri Note that homogeneous points map to the same Cartesian point " # % & and " # % & Projec5on Matri = Projec5on transorma5on: or in 3D coordinates (divide b the 4 th coordinate): =,, We call this division step the perspec5ve divide

25 2/26/3 Perspec5ve Projec5on Maps points onto vie plane along projectors emana5ng rom center o projec5on (COP)! # & &! # # & # & = # & # & # & # & " % # & # & " % Perspec5ve Effects Distant object becomes small The distor5on o items hen vieed at an angle (spa5al oreshortening) 25

26 2/26/3 Proper5es o Perspec5ve Projec5on Perspec5ve projec5on is an eample o projec5ve transorma5on lines maps to lines parallel lines do not necessar remain parallel ra5os are not preserved One advantage o perspec5ve projec5on is that sie varies inversel propor5onal to the distance looks realis5c Perspec5ve Projec5on 26

27 2/26/3 Perspec5ve Projec5on Pietro Perugino (48-82) Mul5ple Centers o Projec5on Creates more realis5c images: Note ho parallel lines in 3D space ma appear to converge to a single point hen vieed in perspec5ve Also called central projec5on: projec5on lines passing through the center (ee point) 27

28 Recall: Orthographic (Parallel) Projec5on Center o Projec5on is at ininit Ras travel parallel to the - ais (orthogonal to the image) World point (,, ) projects to image point (,, ) Orthographic Projec5on Matri = 28

29 Proper5es o Parallel Projec5ons Parallel projec5ons preserve parallelism (parallel lines remain parallel even aqer being larened to 2D) Useul or tech- draing, computer aided design architecture, schema5cs etc. This is because ou can iner the original dimensions o 3D objects rom their 2D images. Orthographic vs. Perspec5ve Object appears same sie, no marer ho ar rom the camera Parallel lines in the orld scene are parallel lines in the image Farther objects appear smaller Parallel lines in the orld scene are not generall parallel lines in the image 29

30 Perspec5ve in OpenGL Perspec5ve in OpenGL Speciing a perspec5ve vie can be done in man as OpenGL supports to methods: glfrustrum and gluperspec5ve 3

31 OpenGL glfrustum glmatrimode(gl_projection); glloadiden5t(); glfrustum(min, ma, min, ma, min, ma); min and ma are speciied as posi5ve distances along - OpenGL gluperspec5ve gluperspec5ve(ov, aspect, near, ar); (ield o vie [ 8]) h 2 θ = tan h = 2near tan θ near 2 2 Onl allos the crea5on o smmetric rustrums. 3

32 gluperspec5ve Parameters Zoom Field o vie: Smaller angle means more oom 32

33 Hands- on Session Run again the projec&on tutor Pla ith projec5on parameters Pla ith camera orienta5on (glulookat) Sitch beteen glfrustum, gluperspec5ve, and glortho Donload robotzoom.cpp and add keboard events to Sitch beteen perspec5ve and orthographic (p/o) Implement transla5on (arros) and oom in /oom out (c/) using glulookat and/or glortho Summar 3D Vieing Camera Posi5oning Projec5on 3D à 2D Orthographic vs. Perspec5ve Projec5on Projec5on Transorma5on Matrices OpenGL Vieing Func5ons glulookat, glortho, glfrustrum, gluperspec5ve 33