CS 450: COMPUTER GRAPHICS

PROJECTIONSSPRING 2015

DR. MICHAEL J. REALE

TRANSFORMS IN REVIEW

• Model and view transform are AFFINE transformations (parallel lines stay parallel)

• Model transform any kind of transformation we’ve seen so far

• Rotation, scaling, translation, shear, etc.

• View transform translation followed by rotation

• Centers camera at origin and orients it to look down –z axis

• Projection transform depends on which we use

• Orthographic affine

• Perspective NOT affine affects w component!

PROJECTION APPROACHES

• One can approach projection in one of two ways:

• Transform x and y, but zero out z component flattening to a 2D plane

• Not invertible (can’t get z information back)

• Doesn’t help us if we’re using z buffers

• Doesn’t restrict which points in z we want (usually want to avoid points BEHIND camera)

• Transform camera volume into unit volume

• Gives us near and far planes limits which range of z coordinates we allow

• Preserves z information (although it does transform it)

• Remember: from here on out, we are assuming that we have performed the view transform and we are already looking down the –z axis

• Either way, output will be normalized device coordinates

ORTHOGRAPHIC PROJECTION

INTRODUCTION

• Orthographic projection

• Parallel lines stay parallel affine transformation

• Does not affect w component

FLATTENING TO A PLANE

• Simple example of flattening to a plane:

1000

0000

0010

0001

oP

USING A VIEW VOLUME

• Define Axis-Aligned Bounding Box (AABB) with 6 values: left, right, bottom, top, near, far

• Goes from (l,b,n) to (r,t,f)

• Points to render must be inside AABB

• Near plane, far plane = extents of z values

• Transform AABB to unit cube

• Translation, then scale

• Makes clipping more efficient

• OpenGL (-1,-1,-1) to (1,1,1)

1000

200

02

0

002

10002

100

2010

2001

1000

02

00

002

0

0002

)()(

nf

nf

nf

bt

bt

bt

lr

lr

lr

nf

bt

lr

nf

bt

lr

tTsSPo

FLIPPING Z

• Technically, because we’re looking down NEGATVE z axis near > far!

• Often switch to LEFT-handed coordinate system by using reflection matrix before projection

• Looking down POSITIVE z axis

• Then, specify near and far values with 0 < n’ < f’ (positive z values)

1000

0100

0010

0001

)1,1,1(S

PERSPECTIVE PROJECTION

INTRODUCTION

• Perspective projection

• NOT affine parallel lines are generally NOT parallel after projection

• Lines may converge to a single point

• Matches more closely how we see things in real life (farther away = looks smaller) more commonly used

FLATTENING TOA PLANE

• Let’s say we have a plane z = -d, d > 0

• Camera is at origin

• We want to project a point p onto the plane z = -d, giving us a new point q = (qx , qy , -d)

• Using similar triangles, we can get the new x and y components:

z

xx

zx

x

p

pdq

p

d

p

q

z

yy

zy

y

p

pdq

p

d

p

q

FLATTENING TO A PLANE• The corresponding perspective projection matrix is given by:

• NOTICE: The last row isn’t just (0,0,0,1) anymore! will alter point’s w coordinate must divide vector by w!

z

xx p

pdq

z

yy p

pdq

0/100

0100

0010

0001

d

Pp

1

/

/

1

/

/

/

/10/100

0100

0010

0001

d

pdp

pdp

pdp

pdp

pdp

dp

p

p

p

p

p

p

d

pPq zy

zx

zz

zy

zx

z

z

y

x

z

y

x

p

USING A VIEW VOLUME

• The view volume (or frustum) of a perspective projection is defined a little differently:

• Near and far plane

• (Left, bottom) and (right, top) corners of NEAR plane

PERSPECTIVE PROJECTION MATRIX

• The perspective projection matrix is given by:

• Example: point already at near plane:

0100

200

02

0

002

nf

fn

nf

nfbt

bt

bt

nlr

lr

lr

n

Pp

1

1

)2(

)2(

1

)(

)2(

)2(

1

)2(

)2(

)2(

)2(

)2(

)2(

2)(

)(2

)(2

10100

200

02

0

002

bt

btplr

lrp

nf

nfbt

btplr

lrp

nf

fnfbt

btplr

lrp

nnf

fnfnbt

btpnlr

lrpn

nnf

fnnfnbt

btnnplr

lrnnp

n

p

p

nf

fn

nf

nfbt

bt

bt

nlr

lr

lr

n

pPq y

x

y

x

y

x

y

x

y

x

y

x

p

FLIPPING Z

• Same as with the orthographic projection, z can be flipped before projection

• After that, use 0 < n’ < f’

• OpenGL matrix with flipping:

0100''

''2

''

''00

0'2

0

00'2

nf

nf

nf

nfbt

bt

bt

nlr

lr

lr

n

POpenGL

FIELD OF VIEW

• Often can define horizontal and vertical field of view (FOV) angles

• Too wide see distortion on edges (fish-eye lens effect)

• Too narrow everything very “zoomed in”

• Can compute horizontal FOV based on w = width of monitor and d = distance from user:

• Asymmetric frustums = where either r != -l or t != -b used in stereo viewing

))2/(arctan(2 dw

EFFECTS OF FOV

FOV = 109°

FOV = 70°

FOV = 30°

EFFECT ON Z VALUE

• With perspective projection resulting z values do NOT vary linearly with input pz value!

• Placement of near and far planes affects precision of z buffer!

• Example: abs(f – n) = 100, varying how close near plane is to camera: