[PPT] - Viewing/Projections IV Week 4, Fri Feb 1 PowerPoint Presentation

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University of British Columbia CPSC 314 Computer Graphics Jan-Apr 2008 Tamara Munzner http://www.ugrad.cs.ubc.ca/~cs314/Vjan2008

Viewing/Projections IV Week 4, Fri Feb 1

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2

News

extra TA office hours in lab next week to

answer questions

Mon 1-3
Tue 2-4
Wed 1-3
reminder
Wed 2/6: Homework 1 due 1pm sharp
Wed 2/6: Project 1 due 6pm.

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3

Review: View Volumes

specifies field-of-view, used for clipping
restricts domain of z stored for visibility test

z perspective view volume perspective view volume

rthographic view volume
rthographic view volume

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

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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

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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Review: Projection Normalization

warp perspective view volume to orthogonal

view volume

render all scenes with orthographic projection!
aka perspective warp

x z=α z=d x z=0 z=d

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Review: 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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Review: Separate Warp From Homogenization

warp requires only standard matrix multiply
distort such that orthographic projection of distorted
bjects 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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Reading for Viewing

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 21.2.2 Visual Perception (Color)

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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

x x y y viewport viewport NDC NDC 500 300

1

1 1

1

height width x x y y

xD yD zD 1

=

1 width 2 1 2 1 height 2 1 2 1 depth 2 1

width

2 height 2 depth 2 1

1

1 1 1

xN

yN zN 1

=

width(xN +1) 1 2 height(yN +1) 1 2 depth(zN +1) 2 1

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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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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(a,x,y,z glRotatef(a,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 framebuffer

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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

x' y' z' w'

=

1 1 1 1/d

x

y z 1

x'

y' z' w'

=

E A F 1 B C D 1

x

y z 1

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Perspective Derivation

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

x' y' z' w'

=

1 1 1 1/d

x

y z 1

x'

y' z' w'

=

E A F 1 B C D 1

x

y z 1

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Perspective Derivation

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

x' y' z' w'

=

1 1 1 1/d

x

y z 1

x'

y' z' w'

=

E A F 1 B C D 1

x

y z 1

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Perspective Derivation

x' y' z' w'

=

E A F B C D 1

x

y z 1

y'= Fy + Bz, y'

w' = Fy + Bz w' , 1= Fy + Bz w' , 1= Fy + Bz z , 1= F y 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= F top near B

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Perspective Derivation

similarly for other 5 planes
6 planes, 6 unknowns

2n r l r + l r l 2n t b t + b t b ( f + n) f n 2 fn f n 1

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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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Perspective Example

view volume

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

2n r l r + l r l 2n t b t + b t b ( f + n) f n 2 fn f n 1

1

1 5/3 8/3 1

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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

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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);

OCS2 OCS2 O2W O2W VCS VCS

modeling modeling transformation transformation viewing viewing transformation transformation projection projection transformation transformation

bject

world viewing W2V W2V V2C V2C WCS WCS

transformations that

are applied first are specified last

OCS1 OCS1 WCS WCS VCS VCS W2O W2O W2O W2O CCS CCS clipping CCS CCS OCS OCS

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Projection Taxonomy

planar planar projections projections perspective: perspective: 1,2,3-point 1,2,3-point parallel parallel

blique
blique
rthographic
rthographic

cabinet cabinet cavalier cavalier top, top, front, front, side side axonometric: axonometric: isometric isometric dimetric dimetric trimetric trimetric

http://ceprofs.tamu.edu/tkramer/ENGR%20111/5.1/20

perspective: projectors converge
orthographic, axonometric: projectors

parallel and perpendicular to projection plane

oblique: projectors parallel, but not

perpendicular to projection plane

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Perspective Projections

projectors converge on image plane
select how many vanishing points
one-point: projection plane parallel to two axes
two-point: projection plane parallel to one axis
three-point: projection plane not parallel to any axis
ne-point
ne-point

perspective perspective two-point two-point perspective perspective three-point three-point perspective perspective

Tuebingen demo: vanishingpoints

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Orthographic Projections

projectors parallel, perpendicular to image plane
image plane normal parallel to one of principal axes
select view: top, front, side
every view has true dimensions, good for measuring

http://www.cs.fit.edu/~wds/classes/cse5255/thesis/images/proj/orthoProj.gif

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Axonometric Projections

projectors parallel, perpendicular to image plane
image plane normal not parallel to axes
select axis lengths
can see many sides at once

http://ceprofs.tamu.edu/tkramer/ENGR%20111/5.1/20

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Oblique Projections

x x y y z z

cavalier

cavalier

d d d d x x y y z z

cabinet

cabinet

d d d / 2 d / 2

projectors parallel, oblique to image plane
select angle between front and z axis
lengths remain constant
both have true front view
cavalier: distance true
cabinet: distance half