Cs543 15 Projection 1up

8/6/2019 Cs543 15 Projection 1up

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CS 543 - Computer Graphics:

Projection

byRobert W. Lindeman

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


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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


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Perspective ProjectionSimilar to real world

Characterized by object foreshorteningObjects appear larger if they are closer to

cameraNeed to defineCenter of projection (COP) Projection (view) plane

ProjectionConnecting the object to the center of

projection

projection plane

camera


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Why is it Called Projection?

View plane


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Orthographic (Parallel) ProjectionNo foreshortening effectDistance from camera does not matter

The center of projection is at infinity

Projection calculation Just choose equal z coordinates


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Field of ViewDetermine how much of the world is

taken into the picture

Larger field of view = smaller object-projection size

x

y

z

y

z

field of view(view angle)

center of projection


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Near and Far Clipping PlanesOnly objects between near and far planes

are drawn

Near 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 window

Objects outside the frustum are clipped

x

y

z

Near plane Far plane

View Frustum


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Projection TransformationIn OpenGL Set the matrix mode to GL_PROJECTION

For perspective projection, use

gluPerspective( fovy, aspect, near, far );or

glFrustum(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


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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


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Example: Projection

Transformation 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 TransformationProjectionMap the object from 3D space to 2D screen

x

y

z

x

y

z

Perspective: gluPerspective( ) Parallel: glOrtho( )


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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 could

just set all Z to the same valuejust set all Z to the same value X

p= 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 domapping from actual view volume to CVV


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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 "(y max+ y min)/20 0 1 "(zmax+ zmin)/2

0 0 0 1

#

$

%

%%%

&

'

(

(((


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Parallel Projection: glOrtho

(cont.)Scaling factor is ratio of cube dimension

to Ortho view volume dimension

Scaling factors

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

So, scaling matrix M2:

2

x max" x min0 0 0

0

2

y max" y min 0 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

x max" x min0 0 0

0 2y max" y min

0 0

0 02

zmax" zmin0

0 0 0 1

#

$

%%%%%%%

&

'

(((((((

X

1 0 0 "(xmax+ xmin)/2

0 1 0 "(y max+ y min)/20 0 1 "(zmax+ zmin)/2

0 0 0 1

#

$

%

%%%

&

'

(

(((


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Perspective Projection: ClassicalSide view

x

y

z

(0,0,0)

d

Projection plane

Eye (center of projection )

(x,y,z)

(x,y,z)

-z

z

y

Based on similar triangles:

y -zy d

dy = 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 example

Q: 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 *

( )= N

Px

"Pz,N

Py

"Pz

#

$%

&

'(


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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+ Py

2+ Pz

2

( )

x*,y*,z*( ) = N Px"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


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Homogenous Coordinates Would like to express projection as 4x4 transformmatrix

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 expressedas (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 / -zy = 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 y

z

#

$% &

'(

"d

1

#

$

%%

%%%

&

'

((

(((


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

Need to modify the projection matrix to

include 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

y

z

Z = 1 z = -1


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Perspective Projection (cont.)Final projection matrix

glFrustum( 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


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ReferencesHill, Chapter 7

Documents

Cs543 15 Projection 1up