Viewing

The Camera and Projection

Gail Carmichael (gail_c@scs.carleton.ca)

The Goal

Understand the process of getting from 3D line

segments to images of these lines on the screen.

Canonical View Volume

Windowing transform brings points to pixels: MW

xpixel

ypixel

1

=

xcanonical

ycanonical

1

Canonical View Volume

Mw

Orthographic Projection

Orthographic Perspective

Orthographic Viewing Volume

Orthographic View to Canonical View

x

y

z

1

World to Canonical Coordinates

Scale Move toOrigin

Orthographic View to Canonical View

2/(r-l)

0 0 0

02

/(t-b)0 0

0 02

/(n-f)0

0 0 0 1

x

y

z

1

1 0 0-(l+r)

/2

0 1 0-(b+t)

/2

0 0 1-(n+f)

/2

0 0 0 1

World to Canonical Coordinates

Drawing Lines in Orthographic View

Mo=Mw Mscale Mmove_to_origin

xpixel

ypixel

zcanonical

1

= Mo

x

y

z

1

Arbitrary View Positions

Camera is looking this

wayCamera is centered here

Top of cameragoes this way

Arbitrary View Positions

w = - (g / ||g||)

u = (t × w) / || t × w ||

v = w × u

Arbitrary View Positions

Coordinate Transformations

Coordinate Transformations

Coordinate Transformations

Coordinate Transformations

p = (xp,yp) ≡ o + xpx + ypy

p = (up,vp) ≡ e + upu + vpv

Coordinate Transformationsp = (xp,yp) ≡ o + xpx + ypy

p = (up,vp) ≡ e + upu + vpv

Coordinate Transformations

xp

yp

1

=

up

vp

1

? ?

p = (xp,yp) ≡ o + xpx + ypy

p = (up,vp) ≡ e + upu + vpv

Coordinate Transformations

xp

yp

1

=

1 0 xe

0 1 ye

0 0 1

up

vp

1

xu xv 0

yu yv 0

0 0 1

p = (xp,yp) ≡ o + xpx + ypy

p = (up,vp) ≡ e + upu + vpv

Camera Coordinate Transform

Camera Coordinate Transform

Mv =

1 0 0 -xe

0 1 0 -ye

0 0 1 -ze

0 0 0 1

xu yu zu 0

xv yv zv 0

xw yw zw 0

0 0 0 1

Drawing with Arbitrary View and Orthographic Projection

xpixel

ypixel

zcanonical

1

= Mo Mv

x

y

z

1

Perspective Projection

ys = y(d/z)

Perspective Via Orthographic

Perspective Via Orthographic

Perspective Via Orthographic

Perspective Transform

Mp =

1 0 0 0

0 1 0 0

0 0(n+f)

/n-f

0 01

/n0

Perspective Transform

Mp

x

y

z

1

=

x

y

z[(n+f)/n] - f

z/n

nx/z

ny/z

n + f – (fn/z)

1

Perspective Transform

Mp =

n 0 0 0

0 n 0 0

0 0 (n+f) -fn

0 0 1 0

Drawing with Arbitrary View and Perspective Projection

xpixel

ypixel

zcanonical

1

= Mo Mp Mv

x

y

z

1

CAUTION!!

Everything up until now used the more common right-hand

coordinate system.

Direct3D uses the left-hand coordinate system.

See:http://msdn.microsoft.com/en-us/library/windows/desktop/bb204853%28v=vs.85%29.aspx