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Einführung in Visual Computing186.822

Viewing

Werner Purgathofer

Viewing in the Rendering Pipeline

Werner Purgathofer 2raster image in pixel coordinates

clipping + homogenization

object capture/creation

modelingviewing

projection

rasterizationviewport transformation

shading

vertex stage(„vertex shader“)

pixel stage(„fragment shader“)

scene in normalized device coordinates

transformed vertices in clip space

scene objects in object space

From Object Space to Screen Space

Werner Purgathofer 3

object space world space camera space

clip space pixel space

„view frustum“modeling

transformationcamera

transformation

projectiontransformation

viewporttransformation

Viewport Transformation







transformation




assumption: scene is in clip space !clip space = [–1,1] × [–1,1] × [–1,1]orthographic camera looking in –z directionscreen resolution nx × ny pixels

Werner Purgathofer / Einf. in Visual Computing 6

nx ny(–1,–1,–1)

(1,1,1)

(–1,–1) (0,0)(1,1) (nx,ny)

clip space:

z


can be done with the matrix

this ignores the z-coordinate, but…

(–1,–1) (0,0)(1,1) (nx,ny)

xscreenyscreen

nx/2 00 ny/2

nx/2ny/2

0 0 11

xy1

= ·

Werner Purgathofer / Einf. in Visual Computing

Projection Transformation







transformation



Parallel Projection (Orthographic Projection)

11

3 parallel-projection views of an object, showing relative proportions from different viewing positions

top side front


Parallel vs. Perspective Projection

12Werner Purgathofer / Einf. in Visual Computing

For Now: Parallel (Orthographic) Projection



clip space screen space



transformation



FB

Projection Transformation (Orthographic)

assumption: scene in box [L,R] × [B,T] × [F,N]orthographic camera looking in –z directiontransformation to clip space

14

L

(L,B,F) (–1,–1,–1)(R,T,N) (1,1,1)

clip space:

(–1,–1,–1)

(1,1,1)

z(L,B,F)

(R,T,N)z

orthographic view volumein camera space:

R

T

N


Projection Transformation (Orthographic)

15

(L,B,F) (–1,–1,–1)(R,T,N) (1,1,1)

0 0 0 1

R – L2

T – B2

N – F2

R – LR + L

T – BT + B

N – FN + F

0 0

0 0

0 0

–

–

–Morth =

1

L

F

B

1

–1

–1

–1· =


Parallel Projection Types

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orthographicprojection

obliqueprojection

different orientation of the projection vector –g

–g

viewing plane

–g

viewing plane


Camera Transformation







transformation



Viewing: Projection Plane

19

coordinate reference for obtaining a selected view of a 3D scene

displayplane


Viewing: Camera Definitionsimilar to taking a photographinvolves selection of

camera positioncamera directioncamera orientation“window” (zoom) of camera

20

worldcoordinates

cameracoordinates


Viewing: Camera Transformation (1)

view reference pointorigin of camera coordinate systemgaze direction or look-at point

21

right-handed camera-coordinate system, with axes u, v, w, relative to world-coordinate scene

y

zx

vw

u

eo



22

eg

e … eye positiong … gaze direction (positive w-axis points to the viewer)t … view-up vector

wu

t

v = w × u

w = – g|g|

u =|t × w|t × w



23

eg

e … eye positiong … gaze direction (positive w-axis points to the viewer)t … view-up vector

vw

ut

v = w × u

w = – g|g|

u =|t × w|t × w



24

aligning viewing system with world-coordinate axes using translate-rotate transformations

Mcam = Rz· Ry· Rx·T

x

y

z

o

uvw

e

x

y

z

uv

wx

y

z

ouv

we








transformation



Viewing: Camera + Projection + Viewport

27

xscreenyscreen

z1

= Mvp·Morth·Mcam ·

xyz1

pixels onthe screen

world coordinates

( )








transformation



Perspective Projection

Perspective Projection

30

perspective projection of equal-sized objects at different distances from the view plane

x

y

zview plane

projection reference

point


Parallel vs. Perspective Projection

31

parallel projection: preserves relative proportions & parallel features (affine transform.)

perspective projection: center of projection, realistic views


Perspective Transform

32

N F

T

B

O


Perspective Transformation

33

fypO

view plane

yp = yfz

z

y

f … focal length

derivation of perspective

transformation


yp fzy =


34

fz

yp

y

view plane

yp = yfz

(z=N)

O(z=0)

N

N N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

= P

xp = xdzNanalogous:

derivation of perspective

transformation



35

N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

xyz1

· =

x·Ny·N

z·(N+F) –F·Nz

~>

x·N/zy·N/z

(N+F) –F·N/z1

homogenization: divide by z

yp = yz

xp = xzN

N

P =derivation of

perspective transformation


Example: (Right) Top Near Corner

36

N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

RTN1

· =

R·NT·N

N·(N+F) –F·NN

~>

RTN1

x (B or T or L or R)

view plane

NO

F


Example: (Left) Top Far Corner

37

N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

L·F/NT·F/N

F1

· =

L·FT·F

F·(N+F) –F·NF

~>

LTF1

x (B or T or L or R)

NO

F

view plane x·F/N


Perspective Transform

38

N F

T

B

O


Nonlinear z-Behaviour








transformation



Viewing: Camera + Projection + Viewport

42

xscreenyscreen

z´1

= Mvp·Morth·P·Mcam ·

xyz1

( )

Mper

pixels onthe screen

world coordinates








transformation



z-Values Remain in Order

44Werner Purgathofer

N 0 0 00 N 0 00 0 N+F –F·N0 0 1 0

xyz1

· ~>

x·N/zy·N/z

(N+F) –F·N/z1

z1, z2, N, F < 0z1 < z2

1/z1 > 1/z2 | · (–F·N) (<0)– F·N/z1 < – F·N/z2 | + (N+F)(N+F) – F·N/z1 < (N+F) – F·N/z2


Perspective Projection Properties

parallel lines parallel to view plane ⇒ parallel linesparallel lines not parallel to view plane ⇒ converging lines (vanishing point)lines parallel to coordinate axis ⇒principal vanishing point (one, two or three)


Principle Vanishing Points

46

2-point perspectiveprojection

3-point perspectiveprojection

1-pointperspectiveprojection

vanishing point


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