[PPT] - Viewing/Projections III Week 4, Wed Jan 31 PowerPoint Presentation

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University of British Columbia CPSC 314 Computer Graphics Jan-Apr 2007 Tamara Munzner http://www.ugrad.cs.ubc.ca/~cs314/Vjan2007

Viewing/Projections III Week 4, Wed Jan 31

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News

extra TA coverage in lab to answer questions
Wed 2-3:30
Thu 12:30-2
my office hours reminder (in lab also)
Wed (today) 11-12
Fri 11-12

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Reading for Today

FCG Chapter 7 Viewing
FCG Section 6.3.1 Windowing Transforms
RB rest of Chap Viewing
RB rest of App Homogeneous Coords

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Reading for Next Time

RB Chap Color
FCG Sections 3.2-3.3
FCG Chap 20 Color
FCG Chap 21 Visual Perception

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Review: Graphics Cameras

real pinhole camera: image inverted

image image plane plane eye eye point point

ν computer graphics camera: convenient equivalent

image image plane plane eye eye point point center of center of projection projection

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Review: Basic Perspective Projection

similar triangles similar triangles

→ = z y d y'

z d y y ⋅ = '

z z P(x,y,z) P(x,y,z) P( P(x x’ ’,y ,y’ ’, ,z z’ ’) ) z z’ ’=d =d y y

z d x x ⋅ = ' d z = '

          1 1 1 1 d

                  d d z y d z x / /

          d z z y x /

homogeneous homogeneous coords coords

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Review: Orthographic Cameras

center of projection at infinity
no perspective convergence
just throw away z values

                        =             1 1 1 1 1 z y x z y x

p p p

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Review: Transforming View Volumes

x z

NDCS

y

(-1,-1,-1) (1,1,1)

rthographic view volume
rthographic view volume

x z VCS y x=left y=top x=right z=-far z=-near y=bottom perspective view volume perspective view volume x=left x=right y=top y=bottom z=-near z=-far x VCS y

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Orthographic Derivation

scale, translate, reflect for new coord sys

x z

VCS

y x=left y=top x=right z=-far z=-near y=bottom x z

NDCS

y

(-1,-1,-1) (1,1,1)

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Orthographic Derivation

scale, translate, reflect for new coord sys

x z

VCS

y x=left y=top x=right z=-far z=-near y=bottom x z

NDCS

y

(-1,-1,-1) (1,1,1)

b y a y + ⋅ = ' 1 ' 1 ' − = → = = → = y bot y y top y

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Orthographic Derivation

scale, translate, reflect for new coord sys

b y a y + ⋅ = ' 1 ' 1 ' − = → = = → = y bot y y top y

bot top bot top b bot top top bot top b bot top top b b top bot top − − − = − ⋅ − − = − ⋅ − = + − = 2 ) ( 2 1 2 1

b bot a b top a + ⋅ = − + ⋅ = 1 1

bot top a top bot a top a bot a bot a top a bot a b top a b − = + − = ⋅ − − ⋅ − = − − ⋅ − − = ⋅ − ⋅ − − = ⋅ − = 2 ) ( 2 ) ( ) 1 ( 1 1 1 1 , 1

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Orthographic Derivation

scale, translate, reflect for new coord sys

x z

VCS

y x=left y=top x=right z=-far z=-near y=bottom

b y a y + ⋅ = ' 1 ' 1 ' − = → = = → = y bot y y top y bot top bot top b bot top a − + − = − = 2

same idea for right/left, far/near same idea for right/left, far/near

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Orthographic Derivation

scale, translate, reflect for new coord sys

P near far near far near far bot top bot top bot top left right left right left right P                         − + − − − − + − − − + − − = 1 2 2 2 '

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Orthographic Derivation

scale, translate, reflect for new coord sys

P near far near far near far bot top bot top bot top left right left right left right P                         − + − − − − + − − − + − − = 1 2 2 2 '

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Orthographic Derivation

scale, translate, reflect for new coord sys

P near far near far near far bot top bot top bot top left right left right left right P                         − + − − − − + − − − + − − = 1 2 2 2 '

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Orthographic Derivation

scale, translate, reflect for new coord sys

P near far near far near far bot top bot top bot top left right left right left right P                         − + − − − − + − − − + − − = 1 2 2 2 '

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Orthographic OpenGL

glMatrixMode glMatrixMode(GL_PROJECTION); (GL_PROJECTION); glLoadIdentity glLoadIdentity(); (); glOrtho glOrtho(left,right, (left,right,bot bot,top,near,far); ,top,near,far);

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Demo

Brown applets: viewing techniques
parallel/orthographic cameras
projection cameras
http://www.cs.brown.edu/exploratories/freeSoftware/catalogs

/viewing_techniques.html

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Projections II

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Asymmetric Frusta

our formulation allows asymmetry
why bother?
z
z

x x Frustum Frustum

right right left left

z
z

x x Frustum Frustum z=-n z=-n z=-f z=-f

right right left left

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Asymmetric Frusta

our formulation allows asymmetry
why bother? binocular stereo
view vector not perpendicular to view plane

Right Eye Right Eye Left Eye Left Eye

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Simpler Formulation

left, right, bottom, top, near, far
nonintuitive
often overkill
look through window center
symmetric frustum
constraints
left = -right, bottom = -top

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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

x x Frustum Frustum z=-n z=-n z=-f z=-f α α

fovx fovx/2 /2 fovy fovy/2 /2

h h w w

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Perspective OpenGL

glMatrixMode(GL_PROJECTION); glLoadIdentity(); glFrustum(left,right,bot,top,near,far);

r

glPerspective(fovy,aspect,near,far);

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Demo: Frustum vs. FOV

Nate Robins tutorial (take 2):
http://www.xmission.com/~nate/tutors.html

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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

OCS OCS O2W O2W VCS VCS CCS CCS NDCS NDCS DCS DCS

modeling modeling transformation transformation viewing viewing transformation transformation projection projection transformation transformation viewport viewport transformation transformation perspective perspective divide divide

bject

world viewing device normalized device clipping W2V W2V V2C V2C N2D N2D C2N C2N WCS WCS

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Projection Normalization

warp perspective view volume to orthogonal

view volume

render all scenes with orthographic projection!
aka perspective warp

Z

x z=α z=d

Z

x z=0 z=d

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Perspective Normalization

perspective viewing frustum transformed to

cube

orthographic rendering of cube produces same

image as perspective rendering of original frustum

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Predistortion

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Demos

Tuebingen applets from Frank Hanisch
http://www.gris.uni-tuebingen.de/projects/grdev/doc/html/etc/

AppletIndex.html#Transformationen

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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

OCS OCS O2W O2W VCS VCS CCS CCS NDCS NDCS DCS DCS

modeling modeling transformation transformation viewing viewing transformation transformation projection projection transformation transformation viewport viewport transformation transformation perspective perspective divide divide

bject

world viewing device normalized device clipping W2V W2V V2C V2C N2D N2D C2N C2N WCS WCS

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Separate Warp From Homogenization

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

distorted objects is desired persp projection

w is changed
clip after warp, before divide
division by w: homogenization

CCS CCS NDCS NDCS

alter w alter w / w / w

VCS VCS

projection projection transformation transformation

viewing normalized device clipping

perspective perspective division division

V2C V2C C2N C2N

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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

            1 z y x             − z z y x

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Perspective Divide Example

            − − − ≡             − =             ⋅             − =                           1 1 / / 1 1 1 1 1 1 z y z x z z y x z y x z y x T

alter w alter w / w / w projection projection transformation transformation perspective perspective division division

after homogenizing, once again w=1

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Perspective Normalization

matrix formulation
warp and homogenization both preserve

relative depth (z coordinate)

                      − ⋅ − =                             − ⋅ − − d z d d z y x z y x d d d d d α α α α α ) ( 1 1 1 1                       − − =           z d d d z y d z x z y x

p p p

α α 1 / /

2

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Demo

Brown applets: viewing techniques
parallel/orthographic cameras
projection cameras
http://www.cs.brown.edu/exploratories/freeSoftware/catalogs

/viewing_techniques.html

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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

OCS OCS O2W O2W VCS VCS CCS CCS NDCS NDCS DCS DCS

modeling modeling transformation transformation viewing viewing transformation transformation projection projection transformation transformation viewport viewport transformation transformation perspective perspective divide divide

bject

world viewing device normalized device clipping W2V W2V V2C V2C N2D N2D C2N C2N WCS WCS

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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

x x y y viewport viewport NDC NDC 500 300

1

1 1

1

x x y y

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

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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

x x y y viewport viewport NDC NDC 500 300

1

1 1

1

x x y y

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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) x x y y viewport viewport NDC NDC 500 300

1

1 1

1

height width x x y y

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N2D Transformation

                  + − + − − + =                         −                                     − − =             1 2 ) 1 ( 2 1 ) 1 ( 2 1 ) 1 ( 1 1 1 1 1 1 2 2 2 1 2 1 2 1 2 1 2 1 2 1 1

N N N N N N D D D

z depth y height x width z y x depth height width depth height width z y x

x x y y viewport viewport NDC NDC 500 300

1

1 1

1

height width x x y y

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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...

x x display display viewport viewport 1024 768 300 500 display height display width x offset y offset y y viewport viewport x x y y

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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 z VCS top view

1
1

1 1

1

NDCS (z not shown) real midpoint xmax-1 DCS (z not shown) ymax-1 x=-1 x=1

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Projective Rendering Pipeline

OCS - object coordinate system WCS - world coordinate system VCS - viewing coordinate system CCS - clipping coordinate system NDCS - normalized device coordinate system DCS - device coordinate system

OCS OCS WCS WCS VCS VCS CCS CCS NDCS NDCS DCS DCS

modeling modeling transformation transformation viewing viewing transformation transformation projection projection transformation transformation viewport viewport transformation transformation alter w alter w / w / w

bject

world viewing device normalized device clipping

perspective perspective division division glVertex3f(x,y,z) glVertex3f(x,y,z) glTranslatef glTranslatef(x,y,z) (x,y,z) glRotatef glRotatef( (th th,x,y,z) ,x,y,z) .... .... gluLookAt gluLookAt(...) (...) glFrustum glFrustum(...) (...) glutInitWindowSize glutInitWindowSize(w,h) (w,h) glViewport glViewport(x,y,a,b) (x,y,a,b)

O2W O2W W2V W2V V2C V2C N2D N2D C2N C2N

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Coordinate Systems

viewing (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 projection matrix

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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=-far x

VCS

y z

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Perspective Derivation

simple example earlier: simple example earlier: complete: shear, scale, projection-normalization complete: shear, scale, projection-normalization

                        − =             1 1 ' ' ' ' z y x D C B F A E w z y x                         =             1 / 1 1 1 1 ' ' ' ' z y x d w z y x

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Perspective Derivation

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

                        − =             1 1 ' ' ' ' z y x D C B F A E w z y x                         =             1 / 1 1 1 1 ' ' ' ' z y x d w z y x

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Perspective Derivation

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

                        − =             1 1 ' ' ' ' z y x D C B F A E w z y x                         =             1 / 1 1 1 1 ' ' ' ' z y x d w z y x

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Perspective Derivation

                        − =             1 1 ' ' ' ' z y x D C B F A E w z y x

z w D Cz z Bz Fy y Az Ex x − = + = + = + = ' ' ' '

, ) ( 1 , 1 , 1 , 1 , ' 1 , ' ' ' , ' B near top F B z y F z z B z y F z Bz Fy w Bz Fy w Bz Fy w y Bz Fy y − − − = − − = − + − = − + = + = + = + = 1 ' / ' 1 ' / ' 1 ' / ' 1 ' / ' 1 ' / ' 1 ' / ' − = → − = = → − = − = → = = → = − = → = = → = w z far z w z near z w y bottom y w y top y w x right x w x left x B near top F − = 1

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Perspective Derivation

similarly for other 5 planes
6 planes, 6 unknowns

                  − − − − + − − + − − + − 1 2 ) ( 2 2 n f fn n f n f b t b t b t n l r l r l r n

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Perspective Example

                  − − − − + − − + − − + − 1 2 ) ( 2 2 n f fn n f n f b t b t b t n l r l r l r n

view volume

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

            − − − 1 3 / 8 3 / 5 1 1

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Perspective Example

            −             − − − =             − − − − 1 1 1 1 3 / 8 3 / 5 1 1 3 / 8 3 / 5 1 1

VCS VCS VCS

z z z

VCS NDCS VCS NDCS VCS NDCS

z z z y z x 3 8 3 5 / 1 / 1 + = = − =

/ w / w

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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);