viewing/projection iv week 4, fri jan 29

University of British ColumbiaCPSC 314 Computer Graphics

Jan-Apr 2010

Tamara Munzner

http://www.ugrad.cs.ubc.ca/~cs314/Vjan2010

Viewing/Projection IV

Week 4, Fri Jan 29

2

News

• extra TA office hours in lab 005• Fri 2-4 (Garrett)

• Tamara's usual office hours in lab• Fri 4-5

• hand in H1 here/now or in box next to 005 labby 5pm• correction: problem 6 worth 54 not 60 marks

3

Review: Basic Perspective Projection

similar trianglessimilar triangles

zz

P(x,y,z)P(x,y,z)

P(x’,y’,z’)P(x’,y’,z’)

z’=dz’=d

yy

homogeneoushomogeneouscoordscoords

€

y'

d=

y

z→ y '=

y ⋅dz

€

x'=x ⋅d

z

€

z'= d

€

x

z /dy

z /d

d

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

€

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1/d 1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

x

y

z

z /d

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

4

Review: View Volumes

• specifies field-of-view, used for clipping

• restricts domain of z stored for visibility test

z

perspective view volumeperspective view volume orthographic view volumeorthographic view volume

x=left

x=right

y=top

y=bottom z=-near z=-farxVCS x

z

VCS

yy

x=lefty=top

x=right

z=-far

z=-neary=bottom

5

Review: Understanding Z

• z axis flip changes coord system handedness• RHS before projection (eye/view coords)

• LHS after projection (clip, norm device coords)

x

z

VCS

yx=left

y=top

x=right

z=-far

z=-neary=bottom

x

z

NDCS

y

(-1,-1,-1)

(1,1,1)

6

Review: Orthographic Derivation

• scale, translate, reflect for new coord sys

x

z

VCS

yx=left

y=top

x=right

z=-far

z=-neary=bottom

x

z

NDCS

y

(-1,-1,-1)

(1,1,1)

€

′ x

′ y

′ z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

e 0 0 f

0 a 0 b

0 0 c d

0 0 0 1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

byay +⋅='

1'

1'

−=→==→=

ybotyytopy

bottop

bottopb

bottopa

−+

−=

−=

2

7

Review: Orthographic Derivation

• scale, translate, reflect for new coord sys

P

nearfar

nearfar

nearfar

bottop

bottop

bottop

leftright

leftright

leftright

P

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−+

−−−

−+

−−

−+

−−

=

1000

200

02

0

002

'

8

Demo

• Robins demo: projection• orthographic

• perspective

9

Projections II

10

Asymmetric Frusta

• our formulation allows asymmetry• why bother?

-z-z

xx

FrustumFrustumrightright

leftleft-z-z

xx

FrustumFrustum

z=-nz=-n z=-fz=-f

rightright

leftleft

11

Asymmetric Frusta

• our formulation allows asymmetry• why bother? binocular stereo

• view vector not perpendicular to view plane

Right EyeRight Eye

Left EyeLeft Eye

12

Simpler Formulation

• left, right, bottom, top, near, far• nonintuitive

• often overkill

• look through window center• symmetric frustum

• constraints• left = -right, bottom = -top

13

Field-of-View Formulation

• FOV in one direction + aspect ratio (w/h)• determines FOV in other direction

• also set near, far (reasonably intuitive)

-z-z

xx

FrustumFrustum

z=-nz=-n z=-fz=-f

fovx/2fovx/2

fovy/2fovy/2hh

ww

14

Perspective OpenGL

glMatrixMode(GL_PROJECTION);glLoadIdentity();

glFrustum(left,right,bot,top,near,far); orglPerspective(fovy,aspect,near,far);

15

Demo: Frustum vs. FOV

• Nate Robins tutorial (take 2): projection• frustum vs perspective

16

Projective Rendering Pipeline

OCS - object/model coordinate system

WCS - world coordinate system

VCS - viewing/camera/eye coordinate system

CCS - clipping coordinate system

NDCS - normalized device coordinate system

DCS - device/display/screen coordinate system

OCSOCS O2WO2W VCSVCS

CCSCCS

NDCSNDCS

DCSDCS

modelingmodelingtransformationtransformation

viewingviewingtransformationtransformation

projectionprojectiontransformationtransformation

viewportviewporttransformationtransformation

perspectiveperspectivedividedivide

object world viewing

device

normalizeddevice

clipping

W2VW2V V2CV2C

N2DN2D

C2NC2N

WCSWCS

17

Perspective Warp

• warp perspective view volume to orthogonal view volume

• render all scenes with orthographic projection!• aka perspective normalization

Z

x

z= z=d

Z

x

z=0 z=d

18

Perspective Warp

• perspective viewing frustum transformed to cube

• orthographic rendering of warped objects in cube produces same image as perspective rendering of original frustum

19

Predistortion

20









CCSCCS

NDCSNDCS

DCSDCS







device

normalizeddevice

clipping

W2VW2V V2CV2C

N2DN2D

C2NC2N

WCSWCS

21

Separate Warp From Homogenization

• warp requires only standard matrix multiply• distort such that orthographic projection of distorted

objects shows desired perspective projection• w is changed

• clip after warp, before divide• division by w: homogenization

CCSCCSNDCSNDCS

alter walter w / w/ w

VCSVCSprojectionprojection

transformationtransformation

viewing normalizeddevice

clipping

perspectiveperspectivedivisiondivision

V2CV2C C2NC2N

22

Perspective Divide Example

• specific example• assume image plane at z = -1

• a point [x,y,z,1]T projects to [-x/z,-y/z,-z/z,1]T

[x,y,z,-z]T

-z-z

€

x

y

z

−z

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

23

Perspective Divide Example

alter walter w / w/ w



• after homogenizing, once again w=1

€

T

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 −1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

⋅

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

x

y

z

−z

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

−x /z

−y /z

−1

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

Perspective Normalization

• matrix formulation

• warp and homogenization both preserve relative depth (z coordinate)€

1 0 0 0

0 1 0 0

0 0d

d − a

−a ⋅dd − a

0 01

d0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

⋅

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

x

y(z − a) ⋅d

d − az

d

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

€

x p

y p

zp

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

x

z /dy

z /dd2

d − a1−

a

z

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

25

Demo

• Brown applets: viewing techniques• parallel/orthographic cameras

• projection cameras

• http://www.cs.brown.edu/exploratories/freeSoftware/catalogs/viewing_techniques.html

26

Perspective To NDCS Derivation

x

z

NDCS

y

(-1,-1,-1)

(1,1,1)x=left

x=right

y=top

y=bottom z=-near z=-farx

VCS

y

z

27

Perspective Derivation

simple example earlier:simple example earlier:

complete: shear, scale, projection-normalizationcomplete: shear, scale, projection-normalization€

x '

y '

z'

w'

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1/d 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

′ x

′ y

′ z

′ w

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

E 0 A 0

0 F B 0

0 0 C D

0 0 −1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

28


earlier:earlier:

complete: shear, scale, projection-normalizationcomplete: shear, scale, projection-normalization€

x '

y '

z'

w'

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1/d 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

′ x

′ y

′ z

′ w

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

E 0 A 0

0 F B 0

0 0 C D

0 0 −1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

29


earlier:earlier:

complete: shear, scale, projection-normalizationcomplete: shear, scale, projection-normalization

€

x '

y '

z'

w'

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1/d 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

′ x

′ y

′ z

′ w

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

E 0 A 0

0 F B 0

0 0 C D

0 0 −1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

30


€

x '

y '

z'

w'

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

E 0 A 0

0 F B 0

0 0 C D

0 0 −1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

x

y

z

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

y'= Fy + Bz, y'

w'=

Fy + Bz

w', 1 =

Fy + Bz

w', 1 =

Fy + Bz

−z,

1 = Fy

−z+ B

z

−z, 1 = F

y

−z− B, 1 = F

top

−(−near)− B,

€

x'= Ex + Az

y'= Fy + Bz

z'= Cz + D

w'= −z

€

x = left → x' /w'=1

x = right → x ' /w'= −1

y = top → y ' /w'=1

y = bottom → y' /w'= −1

z = −near → z' /w'=1

z = − far → z' /w'= −1

€

1 = Ftop

near− B

31


• similarly for other 5 planes

• 6 planes, 6 unknowns

€

2n

r − l0

r + l

r − l0

02n

t − b

t + b

t − b0

0 0−( f + n)

f − n

−2 fn

f − n0 0 −1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

32









CCSCCS

NDCSNDCS

DCSDCS







device

normalizeddevice

clipping

W2VW2V V2CV2C

N2DN2D

C2NC2N

WCSWCS

33

NDC to Device Transformation

• map from NDC to pixel coordinates on display• NDC range is x = -1...1, y = -1...1, z = -1...1

• typical display range: x = 0...500, y = 0...300• maximum is size of actual screen• z range max and default is (0, 1), use later for visibility

xxyy

viewportviewportNDCNDC

0 500

300

0

-1

1

1

-1

xx

yy

glViewport(0,0,w,h);glDepthRange(0,1); // depth = 1 by default

34

Origin Location

• yet more (possibly confusing) conventions• OpenGL origin: lower left

• most window systems origin: upper left

• then must reflect in y• when interpreting mouse position, have to flip your

y coordinates

xxyy


0 500

300

0

-1

1

1

-1

xx

yy

35

N2D Transformation

• general formulation• reflect in y for upper vs. lower left origin

• scale by width, height, depth

• translate by width/2, height/2, depth/2• FCG includes additional translation for pixel centers at

(.5, .5) instead of (0,0)

xxyy


0 500

300

0

-1

1

1

-1height

width

xx

yy

36

N2D Transformation

xxyy


0 500

300

0

-1

1

1

-1height

width

xx

yy€

xD

yD

zD

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

1 0 0width

2−

1

2

0 1 0height

2−

1

2

0 0 1depth

20 0 0 1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

width

20 0 0

0height

20 0

0 0depth

20

0 0 0 1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

1 0 0 0

0 −1 0 0

0 0 1 0

0 0 0 1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

xN

yN

zN

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

width(xN +1) −1

2height(−yN +1) −1

2depth(zN +1)

21

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

37

Device vs. Screen Coordinates

• viewport/window location wrt actual display not available within OpenGL• usually don’t care

• use relative information when handling mouse events, not absolute coordinates

• could get actual display height/width, window offsets from OS

• loose use of terms: device, display, window, screen...

xx

displaydisplayviewportviewport

01024

768

0

300

500

displayheight

display width

x offset

y offsetyy

viewportviewport

xx00 yy

38


OCS - object coordinate system


VCS - viewing coordinate system



DCS - device coordinate system

OCSOCS WCSWCS VCSVCS

CCSCCS

NDCSNDCS

DCSDCS





alter walter w

/ w/ w


device

normalizeddevice

clipping


glVertex3f(x,y,z)glVertex3f(x,y,z)

glTranslatef(x,y,z)glTranslatef(x,y,z)glRotatef(a,x,y,z)glRotatef(a,x,y,z)........

gluLookAt(...)gluLookAt(...)

glFrustum(...)glFrustum(...)

glutInitWindowSize(w,h)glutInitWindowSize(w,h)glViewport(x,y,a,b)glViewport(x,y,a,b)

O2WO2W W2VW2V V2CV2C

N2DN2D

C2NC2N

39

Coordinate Systemsviewing

(4-space, W=1)

clipping

(4-space parallelepiped, with COP moved backwards to infinity

normalized device

(3-space parallelepiped)

device

(3-space parallelipiped)

projection matrix

divide by w

scale &

translate

framebuffer

40

Perspective Example

tracks in VCS: left x=-1, y=-1 right x=1, y=-1

view volume left = -1, right = 1 bot = -1, top = 1 near = 1, far = 4

z=-1

z=-4

x

zVCS

top view

-1-1 1

1

-1NDCS

(z not shown)

realmidpoint

0 xmax-10DCS

(z not shown)

ymax-1

x=-1 x=1

41

Perspective Example

view volume• left = -1, right = 1• bot = -1, top = 1• near = 1, far = 4

€

2n

r − l0

r + l

r − l0

02n

t − b

t + b

t − b0

0 0−( f + n)

f − n

−2 fn

f − n0 0 −1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

€

1 0 0 0

0 1 0 0

0 0 −5 /3 −8 /3

0 0 −1 0

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

42

Perspective Example

/ w/ w

€

xNDCS = −1/zVCS

yNDCS =1/zVCS

zNDCS =5

3+

8

3zVCS

€

1

−1

−5zVCS /3− 8 /3

−zVCS

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

=

1

1

−5 /3 −8 /3

−1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

1

−1

zVCS

1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

43

OpenGL Example

glMatrixMode( GL_PROJECTION );

glLoadIdentity();

gluPerspective( 45, 1.0, 0.1, 200.0 );

glMatrixMode( GL_MODELVIEW );

glLoadIdentity();

glTranslatef( 0.0, 0.0, -5.0 );

glPushMatrix()

glTranslate( 4, 4, 0 );

glutSolidTeapot(1);

glPopMatrix();

glTranslate( 2, 2, 0);

glutSolidTeapot(1);OCS2OCS2

O2WO2W VCSVCSmodelingmodeling

transformationtransformationviewingviewing

transformationtransformation


object world viewingW2VW2V V2CV2CWCSWCS

• transformations that are applied to object first are specified last

OCS1OCS1

WCSWCS

VCSVCS

W2OW2O

W2OW2O

CCSCCSclipping

CCSCCS

OCSOCS

44

Reading for Next Time

• RB Chap Color

• FCG Sections 3.2-3.3

• FCG Chap 20 Color

• FCG Chap 21.2.2 Visual Perception (Color)

viewing/projection iv week 4, fri jan 29

Documents

domain of z

w w pro

view formulation fov

perspective normalizationxz

orthogonal view volumerender

view volumesspecifies

orthographic derivation

projection clip