cs543 15 projection

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CS 543 - Computer Graphics:Projection

byRobert W. Lindeman

[email protected](with help from Emmanuel Agu ;-)

R.W. Lindeman - WPI Dept. of Computer Science 2

3D Viewing and View VolumeRecall: 3D viewing set up

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Projection TransformationView volume can have different shapes

Parallel, perspective, isometric

Different types of projection Parallel (orthographic), perspective, etc.

Important to control Projection type: perspective or orthographic,

etc. Field of view and image aspect ratio Near and far clipping planes


Perspective ProjectionSimilar to real worldCharacterized by object foreshortening

Objects appear larger if they are closer tocamera

Need to define Center of projection (COP) Projection (view) plane

Projection Connecting the object to the center of

projection

projection plane

camera

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Why is it Called Projection?

View plane


Orthographic (Parallel) ProjectionNo foreshortening effect

Distance from camera does not matter

The center of projection is at infinityProjection calculation

Just choose equal z coordinates

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Field of ViewDetermine how much of the world is

taken into the pictureLarger field of view = smaller object-

projection size

x

y

z

y

z θ

field of view(view angle)

center of projection


Near and Far Clipping PlanesOnly objects between near and far planes

are drawnNear plane + far plane + field of view =

View Frustum

x

y

z

Near plane Far plane

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View Frustum3D counterpart of 2D-world clip windowObjects outside the frustum are clipped

x

y

z

Near plane Far plane

View Frustum


Projection TransformationIn OpenGL

Set the matrix mode to GL_PROJECTION For perspective projection, use

gluPerspective( fovy, aspect, near, far );

orglFrustum(left, right, bottom, top,

near, far );

For orthographic projection, useglOrtho( left, right, bottom, top,

near, far );

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gluPerspective( fovy, aspect, near, far )

Aspect ratio is used to calculatethe window width

x

y

z

y

z

fovy

eye

near farAspect = w / h

w

h

θ


glFrustum( left, right, bottom, top,near, far )

Can use this function in place ofgluPerspective( )

x

y

z

left

rightbottom

top

near far

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glOrtho( left, right, bottom, top,near, far )

For orthographic projection

x

y

z

left

rightbottom

top

nearfar


Example: ProjectionTransformation

void display( ) { glClear( GL_COLOR_BUFFER_BIT ); glMatrixMode( GL_PROJECTION ); glLoadIdentity( ); gluPerspective( FovY, Aspect, Near, Far ); glMatrixMode( GL_MODELVIEW ); glLoadIdentity( ); gluLookAt( 0, 0, 1, 0, 0, 0, 0, 1, 0 ); myDisplay( ); // your display routine}

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Projection TransformationProjection

Map the object from 3D space to 2D screen

x

y

z

x

y

z

Perspective: gluPerspective( ) Parallel: glOrtho( )


Parallel Projection (The Math) After transforming the object to eye space,After transforming the object to eye space,

parallel projection is relatively easy: we couldparallel projection is relatively easy: we couldjust set all Z to the same valuejust set all Z to the same value Xp = x Yp = y Zp = -d

We actually want to remember Z – why?

x

y

z(x,y,z)

(Xp, Yp)

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Parallel ProjectionOpenGL maps (projects) everything in

the visible volume into a canonical viewvolume (CVV)

(-1, -1, 1)

(1, 1, -1)

Canonical View VolumeglOrtho( xmin, xmax, ymin, ymax, near, far )

(xmin, ymin, near)

(xmax, ymax, far)

Projection: Need to build 4x4 matrix to do mapping from actual view volume to CVV


Parallel Projection: glOrthoParallel projection can be broken down

into two parts Translation, which centers view volume at

origin Scaling, which reduces cuboid of arbitrary

dimensions to canonical cubeDimension 2, centered at origin

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Parallel Projection: glOrtho(cont.) Translation sequence moves midpoint of view

volume to coincide with origin e.g., midpoint of x = (xmax + xmin)/2

Thus, translation factors are-(xmax+xmin)/2, -(ymax+ymin)/2, -(far+near)/2

So, translation matrix M1:

!

1 0 0 "(xmax+ xmin) /2

0 1 0 "(ymax+ ymin) /2

0 0 1 "(zmax+ zmin) /2

0 0 0 1

#

$

% % % %

&

'

( ( ( (


Parallel Projection: glOrtho(cont.)Scaling factor is ratio of cube dimension

to Ortho view volume dimensionScaling factors

2/(xmax-xmin), 2/(ymax-ymin), 2/(zmax-zmin)

So, scaling matrix M2:

!

2

xmax" xmin0 0 0

02

ymax" ymin0 0

0 02

zmax" zmin0

0 0 0 1

#

$

% % % % % % %

&

'

( ( ( ( ( ( (

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Parallel Projection: glOrtho()(cont.)Concatenating M1xM2, we get transform

matrix used by glOrtho

!

M2 "M1=

2 /(xmax# xmin) 0 0 #(xmax+ xmin) /(xmax# xmin)

0 2 /(ymax# ymin) 0 #(ymax+ ymin) /(ymax# ymin)

0 0 2 /(zmax# zmin) #(zmax+ zmin) /(zmax# zmin)

0 0 0 1

$

%

& & & &

'

(

) ) ) )

Refer to: Hill, 7.6.2

!

2

xmax" xmin0 0 0

02

ymax" ymin0 0

0 02

zmax" zmin0

0 0 0 1

#

$

% % % % % % %

&

'

( ( ( ( ( ( (

X

!

1 0 0 "(xmax+ xmin) /2

0 1 0 "(ymax+ ymin) /2

0 0 1 "(zmax+ zmin) /2

0 0 0 1

#

$

% % % %

&

'

( ( ( (


Perspective Projection: ClassicalSide view

x

yz

(0,0,0)

d

Projection plane

Eye (center of projection )

(x,y,z)

(x’,y’,z’)

-z

z

yBased on similar triangles:

y -z y’ d

d y’ = y * -z

=

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Perspective Projection:Classical (cont.) So (x*, y*), the projection of point, (x, y, z)

onto the near plane N, is given as

Similar triangles

Numerical exampleQ: Where on the viewplane does P = (1, 0.5, -

1.5) lie for a near plane at N = 1?

(x*, y*) = (1 x 1/1.5, 1 x 0.5/1.5) = (0.666, 0.333)

!

x*,y *( ) = NPx

"Pz,N

Py

"Pz

#

$ %

&

' (


Pseudo Depth Checking Classical perspective projection drops z coordinates But we need z to find closest object (depth testing) Keeping actual distance of P from eye is

cumbersome and slow

Introduce pseudodepth: all we need is a measureof which objects are further if two points project tothe same (x, y)

Choose a, b so that pseudodepth varies from –1 to1 (canonical cube)

!

distance = Px2

+ Py2

+ Pz2( )

!

x*,y*,z *( ) = NPx

"Pz,N

Py

"Pz,aPz + b

"Pz

#

$ %

&

' (

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Pseudo Depth Checking (cont.)Solving:

For two conditions, z* = -1 when Pz = -Nand z* = 1 when Pz = -F, we can set uptwo simultaneous equations

Solving for a and b, we get

!

z* =aP

z+ b

"Pz

!

a ="(F + N)

F " N

!

b ="2FN

F " N


Homogenous Coordinates Would like to express projection as 4x4 transform

matrix Previously, homogeneous coordinates for the point P =

(Px, Py, Pz) was (Px, Py, Pz, 1) Introduce arbitrary scaling factor, w, so that P = (wPx,

wPy, wPz, w) (Note: w is non-zero) For example, the point P = (2, 4, 6) can be expressed

as (2, 4, 6, 1) or (4, 8, 12, 2) where w=2 or (6, 12, 18, 3) where w = 3

So, to convert from homogeneous back to ordinarycoordinates, divide all four terms by last component anddiscard 4th term

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Perspective ProjectionSame for x, so we have x’ = x * d / -z y’ = y * d / -z z’ = -d

Put in a matrix form

OpenGL assumes d = 1, i.e., the image plane is at z = -1!

1 0 0 0

0 1 0 0

0 0 1 0

0 0 1"d( ) 1

#

$

% % % %

&

'

( ( ( (

x

y

z

1

#

$

% % % %

&

'

( ( ( (

=

x '

y '

z'

w

#

$

% % % %

&

'

( ( ( (

)

"d xz( )

"d yz

# $ % &

' (

"d

1

#

$

% % % % %

&

'

( ( ( ( (


Perspective Projection (cont.)We are not done yet!

Need to modify the projection matrix toinclude a and b

x’ 1 0 0 0 x y’ = 0 1 0 0 y z’ 0 0 a b z w 0 0 (1/-d) 0 1

We have already solved a and b

x

yz

Z = 1 z = -1

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Perspective Projection (cont.)Not done yet! OpenGL also normalizes

the x and y ranges of the view frustumto [-1, 1] (translate and scale)

So, as in ortho, to arrive at finalprojection matrix We translate by

–(xmax + xmin)/2 in x -(ymax + ymin)/2 in y

And scale by2/(xmax – xmin) in x2/(ymax – ymin) in y


Perspective Projection (cont.)Final projection matrixglFrustum( xmin, xmax, ymin, ymax, N, F )

N = near plane, F = far plane

!

2N

xmax" xmin0

xmax+ xmin

xmax" xmin0

02N

ymax" ymin

ymax+ ymin

ymax" ymin0

0 0"(F + N)

F " N

"2FN

F " N0 0 "1 0

#

$

% % % % % % %

&

'

( ( ( ( ( ( (

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Perspective Projection (cont.)After perspective projection, viewing

frustum is also projected into a canonicalview volume (like in parallel projection)

(-1, -1, 1)

(1, 1, -1)

Canonical View Volume

x

y

z


ReferencesHill, Chapter 7

cs543 15 projection

Education

z max z

lindeman wpi

y max y

glortho parallel projection

projection type

x max x min000

perspective projection

computer science13example