CS-378: Game Technology Lecture #2.1: Projection Prof. Okan Arikan University of Texas, Austin Thanks to James O’Brien, Steve Chenney, Zoran Popovic, Jessica

CS-378: Game TechnologyLecture #2.1: Projection

Prof. Okan ArikanUniversity of Texas, Austin

Thanks to James O’Brien, Steve Chenney, Zoran Popovic, Jessica Hodgins

V2005-05-1.3

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Today

Windowing and Viewing Transformations

Windows and viewports

Orthographic projection

Perspective projection

33

Screen Space

Monitor has some number of pixelse.g. 1024 x 768

Some sub-region used for given program

You call it a window

Let’s call it a viewport instead

44

Screen Space

May not really be a “screen”Image file

Printer

Other

Little pixel details

Sometimes oddUpside down

HexagonalFrom Shirley textbook.

55

Screen Space

Viewport is somewhere on screenYou probably don’t care where

Window System likely manages this detail

Sometimes you care exactly where

Viewport has a size in pixelsSometimes you care (images, text, etc.)

Sometimes you don’t (using high-level library)

66

Canonical View Space

Canonical view region2D: [-1,-1] to [+1,+1]

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77


Canonical view region2D: [-1,-1] to [+1,+1]

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y

textb

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88


Canonical view region

2D: [-1,-1] to [+1,+1]

Define arbitrary window and define objects

Transform window to canonical region

Do other things (we’ll see clipping latter)

Transform canonical to screen space

Draw it.

99


World Coordinates Canonical Screen Space

(Meters) (Pixels)

Note distortion issues...

1010

Projection

Process of going from 3D to 2D

Studies throughout history (e.g. painters)

Different types of projectionLinear

Orthographic

Perspective

Nonlinear

Orthographic is special case ofperspective...

Many special cases in books just one of these two...

1111

Linear Projection

Projection onto a planar surface

Projection directions eitherConverge to a point

Are parallel (converge at infinity)

1212

Linear Projection

A 2D view

OrthographicPerspective

1313

Linear Projection

Orthographic Perspective

1414

Linear Projection

Orthographic Perspective

1515

OrthographicPerspective

Note how different things can be seen

Parallel lines ŅmeetÓ at infinity

Linear Projection

A 2D view

1616

Orthographic Projection

No foreshortening

Parallel lines stay parallel

Poor depth cues

1717


Canonical view region3D: [-1,-1,-1] to [+1,+1,+1]

Assume looking down -Z axisRecall that “Z is in your face”

[1,1,1][-1,-1,-1]

-Z

[1,1,1][-1,-1,-1]

-Z

1818


Convert arbitrary view volume to canonical

[1,1,1] [-1,-1,-1]

-Z

[1,1,1] [-1,-1,-1]

-Z

1919


View vector

Up vector

Right = view X up

Origin

Center

near,top,right

far,bottom,left

*Assume up is perpendicular to view.

2020


Step 1: translate center to origin

2121



Step 2: rotate view to -Z and up to +Y

2222




Step 3: center view volume

2323





Step 4: scale to canonical size

2424

Orthographic ProjectionOrthographic Projection





2525

Perspective Projection

Foreshortening: further objects appear smaller

Some parallel line stay parallel, most don’t

Lines still look like lines

Z ordering preserved (where we care)

2626


Pinhole a.k.a center of projection

Image f

rom

D.

Fors

yth

2727


Foreshortening: distant objects appear smaller

Image f

rom

D.

Fors

yth

2828


Vanishing pointsDepend on the scene

Not intrinsic to camera

“One point perspective”

2929



Nor intrinsic to camera

“Two point perspective”

3030



Not intrinsic to camera

“Three point perspective”

3131


u

v

n

u

v

n

View Frustum

3232


ViewUp

Distance to image planei

Y

-Z

Topt

Bottomb

Nearn

Farf

Center

3333


Step 1: Translate center to origin

Y

-Z

Y

-Z

3434



Step 2: Rotate view to -Z, up to +Y

Y

-Z

Y

-Z

3535




Step 3: Shear center-line to -Z axis

Y

-Z

Y

-Z

3636





Step 4: Perspective

-Z-Z

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3737


Step 4: PerspectivePoints at z=-i stay at z=-i

Points at z=-f stay at z=-f

Points at z=0 goto z=±∞

Points at z=-∞ goto z=-(i+f)

x and y values divided by -z/i

Straight lines stay straightDepth ordering preserved in [-i,-f ]

Movement along lines distorted

-Z-Z

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3838

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3939

Perspective Projection“Eye” plane

Top

Near Far

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lines

View vector

4040


Visualizing division of x and y but not z

4141


Motion in x,y

4242


Note that points on near plane fixed

4343


Recall that points on far plane willstay there...

4444


When we also divide z points mustremain on straight lines

4545


Lines extend outside view volume

4646


Motion in z

4747


Motion in z

4848


Motion in z

4949


Total motionTotal motion

5050


Step 1: Translate center to orange



Step 4: Perspective



-Z-Z

5151


Step 1: Translate center to orange



Step 4: Perspective



-Z-Z

5252


There are other ways to set up the projection matrix

View plane at z=0 zero

Looking down another axis

etc...

Functionally equivalent

5353

Vanishing Points

Consider a ray:

dp dp

5454

Vanishing Points

Ignore Z part of matrix

X and Y will give location in image plane

Assume image plane at z=-i

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Vanishing Points

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5656

Vanishing Points

Assume

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5757

Vanishing Points

All lines in direction d converge to same point in the image plane -- the vanishing point

Every point in plane is a v.p. for some set of lines

Lines parallel to image plane ( ) vanish at infinity

y

x

d

d

t

Lim

y

x

d

d

t

Lim

What’s a horizon?

5858

Ray Picking

Pick object by picking point on screen

Compute ray from pixel coordinates.

5959

Ray Picking

Transform from World to Screen is:

Inverse:

What Z value?

w

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6060

Ray Picking

Recall that:Points at z=-i stay at z=-i

Points at z=-f stay at z=-f

Depends on screen details, YMMVGeneral idea should translate...

6161

Suggested Reading

Fundamentals of Computer Graphics by Pete Shirley

Chapter 6

Documents

CS-378: Game Technology Lecture #2.1: Projection Prof. Okan Arikan University of Texas, Austin Thanks to James O’Brien, Steve Chenney, Zoran Popovic, Jessica