3d viewing 2

7/30/2019 3D Viewing 2

1/10

Copyright of most images by Computer Graphics C Version Hearn D. & Baker M.P. Computergraphik @ TU Wien

ThreeThree--Dimensional ViewingDimensional Viewing

overview of 3D viewing conceptsoverview of 3D viewing concepts

3D viewing pipeline3D viewing pipeline

3D viewing3D viewing--coordinate parameterscoordinate parameters

transformation worldtransformation world viewing coordinatesviewing coordinates

projection transformationsprojection transformations

orthogonal and parallel projectionsorthogonal and parallel projections perspective projectionsperspective projections

viewport transformation & 3D screen coord.viewport transformation & 3D screen coord.


3D Display Methods3D Display Methods

coordinate reference for obtaining a

selected view of a 3D scene

display

plane


3D Display: Wireframe Display3D Display: Wireframe Display

wireframe display of 3 objects, with back lines

removed, from a commercial database of object

shapes. Each object

in the database isdefined as a grid of

coordinate points,

which can then be

viewed in wireframe

form or in a surface-

rendered form

7/30/2019 3D Viewing 2

2/10


3D Display: Parallel Projection3D Display: Parallel Projection

3 parallel-projection views of an object,

showing relative proportions from

different viewing positions


3D Display: Perspective Projection3D Display: Perspective Projection


3D Display: Depth Cueing3D Display: Depth Cueing

= or ?

intensity decreases

with increasing

distance


3D Display: Visibility3D Display: Visibility

visible line and surface identificationvisible line and surface identification

= or !

7/30/2019 3D Viewing 2

3/10


3D Display:3D Display:

Depth Cueing + VisibilityDepth Cueing + Visibility

only visible linesonly visible lines

intensity decreasesintensity decreases

with increasing distancewith increasing distance


3D Display: Surface Rendering3D Display: Surface Rendering

realistic room display achieved with stochastic

ray-tracing methods that apply perspective

projection

surface-texture

mapping

illumination

models


Other 3D Display MethodsOther 3D Display Methods

explodedexploded andand

cutaway viewscutaway views

a fully rendered

turbine can also

be viewed as a

surface-rendered

exploded display


exploded andexploded and

cutawaycutaway viewsviews

color-coded cutaway

view of a lawn

mower engine

showing the

structure and

relationship of

internal components

Other 3D Display MethodsOther 3D Display Methods

7/30/2019 3D Viewing 2

4/10


two views (one for the left, one for thetwo views (one for the left, one for the

right eye)right eye) head mounted displays (hmd)head mounted displays (hmd)

raster monitor with (shutter) glassesraster monitor with (shutter) glasses

3D Display: Stereoscopic Views3D Display: Stereoscopic Views


3D Viewing Pipeline3D Viewing Pipeline

general 3-dim. transformation pipeline, from

modeling coordinates to final device coordinates

modelingtransformation

viewingtransformation

projectiontransformation

viewporttransformation

modeling

coord.

world

coord.

device

coord.

projection

coord.

viewing

coord.

normalizationtransformation

and clipping

normalized

coord.


3D Viewing: Camera Definition3D Viewing: Camera Definition

similar to taking a photographsimilar to taking a photograph

involves selection ofinvolves selection of camera positioncamera position camera orientationcamera orientation windowwindow (aperture) of camera(aperture) of camera


3D Viewing Coordinates (1)3D Viewing Coordinates (1)

view reference pointview reference point origin of viewingorigin of viewing--coordinate systemcoordinate system camera position or lookcamera position or look--at pointat point

right-handed

viewing-coord.

system, with

axes x0, y0, z0,

relative to

world-coord.

scene

7/30/2019 3D Viewing 2

5/10



view-plane normal vector N (= positive zv-axis,

points to the viewer)

N=(1,0,0)

N=(1,0,1)



orientation ofthe view plane

for a specified

look-at point P,

relative to the

viewing-

coordinateorigin P0

view-plane normal vector N (positive zv-axis)



choose arbitrary up-vector

and adjust it perpendicular

to normal vector N

often: choose V

along the yw axis

desired direction

choosing the view-up vector V (positive yv-axis)



viewingviewing--coordinate systemcoordinate system u = vu = v xx nn (positive x(positive xvv--axis)axis) viewview--plane distanceplane distance

a right-handed viewing

system defined with unitvectors u, v, and n view-plane positioningalong the zv axis

7/30/2019 3D Viewing 2

6/10



viewing ascene from

different

directions with

a fixed view-

reference

point



moving

around in ascene by

changing the

position of

the view

reference

point



TRRRM xyzVCWC ,aligning viewing system with world-coordinate

axes using translate-rotate transformations


ProjectionsProjections

parallel projection:

parallel lines,

preserves relative

proportions

perspective projection:

center of projection,realistic views

7/30/2019 3D Viewing 2

7/10


Parallel Projection (1)Parallel Projection (1)

orthographicprojection

obliqueprojection

orientation of the projection vectorVp


Parallel Projection (2)Parallel Projection (2)

orthographic projectionsof an object

isometric projectionfor a cube

plan view

side

elevationview

front

elevationview


Orthographic Parallel ProjectionOrthographic Parallel Projection

=

1000

0000

0010

0001

parallelM

xp = x

yp = y

zp = 0


Oblique Parallel ProjectionOblique Parallel Projection

siny

cos

p Ly

Lxxp

++

=

1000

0000

0tansin

10

0tancos

01

parallelM

tan/ /tan zL Lz= =

7/30/2019 3D Viewing 2

8/10


Parallel Proj.: Cavalier ProjectionParallel Proj.: Cavalier Projection

depth of the cube is projected

equal to the width and the heightCopyright of most images by Computer Graphics C Version Hearn D. & Baker M.P. Computergraphik @ TU Wien

Parallel Proj.: Cabinet ProjectionParallel Proj.: Cabinet Projection

depth of the cube is projected as

one-half that of the width and height


Perspective ProjectionPerspective Projection

perspective projection of equal-sized objects

at different distances from the view plane

view plane

projectionreference

point


z=0

11stst Derivation of Perspective (1)Derivation of Perspective (1)

xp : x = dp : (dp z)x

y

zdp

xp

Pp z

x

P(x,y,z)

(0,0,zprp)

xp =

xdp

dpz yp =ydp

dpz zp = 0

1. forzvp = 0

7/30/2019 3D Viewing 2

9/10


z=zvp

11stst Derivation of Perspective (2)Derivation of Perspective (2)

xp : x = dp : (zprp- z)

xp

Pp z

x

P(x,y,z)(0,0,zprp)

2. forzvp 0

xp =xdp

zprpz yp =ydp

zprpz zp = zvp

x

y

z

dp


22ndnd Derivation of Perspective (1)Derivation of Perspective (1)

uzzzz

yuyy

xuxx

prp)(

zz

zzuzz

prp

vp

vp =

perspective

projection of apoint P (x,y,z)

to position

(xp, yp, zp) on

the view plane

(x,y,z) any point on line for0 u 1

u=0 u=1



= zz dyzz zzyy prp pprp vpprpp

=xpvpprpp zzd

zz

zzu

prp

vp =x xu = x (1 u)

zzd

xprp

p

zz

zzx

prp

vpprp=xp


h 1 !!!


p

prp

dzz

h =

hy

hx

h

h

/y

/x

p

p ==

1//100

)/(/00

0010

0001

z

y

x

dzd

dzzdz

h

z

y

x

pprpp

pprpvppvph

h

h

h 1 !!!

7/30/2019 3D Viewing 2

10/10


Perspective Projection PropertiesPerspective Projection Properties

special cases: zspecial cases: zvpvp=0 or z=0 or zprpprp=0=0

parallel lines parallel to view planeparallel lines parallel to view plane

parallel linesparallel lines

parallel lines not parallel to view planeparallel lines not parallel to view plane

converging lines (vanishing point)converging lines (vanishing point)

lines parallel to coordinate axislines parallel to coordinate axis

principal vanishing point (one, two or three)principal vanishing point (one, two or three)


Vanishing PointsVanishing Points

perspective views & principle vanishing pointsof a cube for various orientations of the viewplane relative to the principle axes of the object

1-point persp. proj. 2-point persp. proj.

3d viewing 2

Documents