3d viewing perspective projections single point perspective cop on x-axis cop (-1/p 0 0 1) vp x (1/p...

3D ViewingPerspective Projections

Single Point Perspective

1111

1

1

1000

0100

0010

001

1

***

pxz

pxy

pxx

zyx

pxzyx

p

zyx

COP on X-axis

COP (-1/p 0 0 1) VPx (1/p 0 0 1)

3D ViewingPerspective Projections

Two Point Perspective

1000

0100

010

001

q

p

PPP qppq

3D ViewingPerspective Projections

Three Point Perspective

1000

100

010

001

P P qp

r

q

p

PP rpqr

3D ViewingPerspective Projections

3D ViewingVanishing Points

Two ways• Intersection of transformed lines• Transformation of points at infinity

X

Z

Y

Y

X

VPx

VPz

3D Viewing

Orthographic

Plane Geometric Projections

Parallel Perspective

Axonometric Oblique

Trimetric Dimetric Isometric

Cavalier Cabinet

Single Point

Two Point

Three Point

3D ViewingImplementation Issues

More from Interface point of view

Y

Z

X

Eye

N

V

U

World Coordinate System (WCS)

Viewing Coordinate System (VCS)

3D ViewingView Coordinate System (VCS)

Viewing coordinate system• Position and orientation of the view plane• Extent of the view plane (window)• Position of the eye

View Plane• View Reference Point (VRP): the origin of VCS specified as (rx , ry, rz) in WCS: center of the scene• Normal to the view plane (nx , ny, nz )

3D Viewing

View Plane• Normal Direction (View Plane Normal VPN) n (nx ,ny ,nz)

User may provide normalized vectore.g.

nx = sin cos ny = sin sin

nz = cos

Z

Y

r

X

View Coordinate System (VCS)

3D Viewing

View Plane• Direction vv is a unit vector intuitively corresponding to “up” vector“up” vector is specified by the user in WCS

n

upup’

v

up’ = up – (up.n)n

v = up’ / |up’|

• Direction u

u = n x v ( Left Handed)

View Coordinate System (VCS)

3D Viewing

Window and Eye

• Window : left, right, bottom,top (wl,wr,wb,wt) generally is centered at VRP (origin)

• Eye : e = (eu,ev,en) Typically e = (0,0,-E)

View Coordinate System (VCS)

u

e

wt

wb

wr

wl

n v

3D ViewingTransformation from WCS to VCS

rba

rv

ubayx

M ) (

) () (

(x, y)

X

Y

O

O’

u

v

r

3D ViewingTransformation from WCS to VCS

Point object is represented as • (a,b,c) in VCS• (x,y,z) in WCS

zyx

zyx

zyx

nnn

vvv

uuu

n

v

u

M

3D Viewing

TMrp

Mrpcba

rMcbazyxp

)(

)(

1

Transformation from WCS to VCS

Conversion from one coordinate system to another

Therefore a=(p-r).u, b=(p-r).v, c=(p-r).n

3D Viewing

1???

0

0

0T

wv

MA

In Homogenous Coordinates

(a,b,c,1) = (x,y,z,1) Awv

Transformation from WCS to VCS

3D Viewing

ntranslatio

TTT rMpMMrpcba )(

In Homogenous Coordinates

r’= -rMT = (-r.u,-r.v,-r.n) = (rx’,ry’,rz’)puvn=pxyzAwv

1'''

0

0

0

1'''

0

0

0

zyx

zzz

yyy

xxx

zyx

T

wv

rrr

nvu

nvu

nvu

rrr

MA

Transformation from WCS to VCS

3D ViewingTransformation from VCS to View Plane

et=0

p*

pt=1

t=t’

unv

e

p (pu,pv,pn)

p*(u*,v*)

Parametrically r(t) = e(1-t)+p.t

3D ViewingTransformation from VCS to View Plane

On u-v plane,r(t)n = 0

nn

nvvn

nn

nuun

nn

n

nn

pepepe

v

pepepe

u

pee

t

tpte

*

*

'

'' )()1(0

3D Viewing

1000

100

0010

0001

1

n

p eM

When eye is on n-axis eu=ev=0u*=pu/(en-pn), v*=pv/(en-pn)

Matrix form (n*=0) Perspective Transformation

1000

000

0010

0001

1

ne

Transformation from VCS to View Plane

3D Viewing

Using Perspective Transformation Mp

Transformation from VCS to View Plane

)depth pseudo(

)1,,,(),,(

*

*

*

****

nn

n

nn

v

nn

u

pnvu

pep

n

pep

v

pep

u

Mpppnvup

3D Viewing

1000

01

0010

0001

11

nvnu

s eeeeM

p*=(pu,pv,pn,1)MsMp

q : in WCS maps to p*=qAwvMsMp

Transformation from VCS to View Plane

If eye is off n-axis we have another matrix

3D ViewingView Volume

Back Plane n=B

Front Plane n=F

View Plane, n=0Eye

3D ViewingView Volume

v v

n n

wt

wb

wt

wbF B

F/(1-F/en) B/(1-B/en)

3D ViewingVolume Normalization Transformation

Vt

Vb

Vl Vr

0

1v

u

3D ViewingVolume Normalization Transformation

F/(1-F/en) B/(1-B/en)

no

0

nt

1

For n

)()(

)(

))((

11

1

2 BFeBeF

FBe

nBeFe

eFF

eBB

eFF

n

nn

n

n

onn

nn

n

o

t

Scaling sn Translation rn

3D Viewing

1

000

000

000

nvu

n

v

n

rrr

s

s

s

N )(/)))(((

)/()()/()(

2 FBeFeBes

wwvvswwvvs

nnnn

btbtv

rlrlu

Volume Normalization Transformation

where

)(/)(

)/()()/()(

BFeBeFr

wwwvwvrwwwvwvr

nnn

btbttbv

rlrllru

Total Transformation: AwvMsMpN