Projection- parallel and perspective

8/17/2019 Projection- parallel and perspective

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Projection


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Defnition

“ Projection can be defned as a mapping

o point P(x,y,z) onto its image P’(x’,y’,z’)in the projection plane, which constittethe display srace!" #he mapping isdetermind by a projection line called the

“Projector! that passes throgh P andintersect the $iew plane" #he intersectionpoint is P’"

P(x,y,z)

P’(x’,y’,z’)

Projectorprojection plane


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#axonomy o Projection


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Perspecti$e Projection

Basic Principles% &

“ #he techni'es o

projection are generalizations o theprinciples sed by artists in preparingperspecti$e drawing o D objectsand scenes" #he eye o the artist is

placed at the center o projection,and the can$as ($iew plane)"!


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athematical Description

* Perspecti$e projection isdetermined by prescribing a center oprojection and a $iew plane" #he $iew

plane is determined by its viewreference point +o and view plane

normal " the object point P is located

in world co&ordinates at (x,y,z)" #heproblem is to determine the image point co-ordinates P’(x’,y’,z’)"


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-iew Plane

.enter o projection

-iew Plane

P1(x,y,z)

P2

P’1(x’,y’,z’)

/

0

1

C

-iew Plane ormal


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.haracteristics2*nomalies

3" Perspecti$e 4oreshortening%&

“ #he arther an object is rom the center

o projection, the smaller it appears"!

5xample% &6'are * is larger in size than s'are 7

bt at $anishing point in $iewing plane theyappears

to be o same size"

A BVanishing poin

Viewing plane Y


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8" -anishing Points

• #hese points are ormed by the intersectiono lines parallel to one o the three principal

axis" #he nmber o principal $anishingpoints is determined by the nmber oprincipal axis intersected by the $iew plane"


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" -iew .onsion• 9bjects behind the center o

projection are projected pside downand bac:ward onto the $iew plane"

center o projectio

P1

P3P2

P2’

P1’

P3’

Y

!

19


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;" #opological Distortion#he points o the plane that is parallel to the $iew

plane < also passes throgh the .entre o

projection

are projected to infnity by the perspecti$eprojection"

=hen we join the point which is bac: o the$iewer to

the point which is ront o the $iewer then the line

will be projected as a bro:en line o infnite extent"

1

0

-iewing Plane

*’ at infnity

*

.

7 >4>>#07’ .’ *’

.entreo

projection


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#ypes o -anishing Points

• 9ne Principal -anishing Point Perspecti$eProjection

• #wo Principal -anishing Point Perspecti$eProjection

• #hree Principal -anishing Point Perspecti$Projection

9 P i i l - i hi P i


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9ne Principal -anishing PointPerspecti$e Projection

• #his Perspecti$e projection occrs when

the projection plane is perpendiclar toone o the Principal axis (x or y or z)"

• *ssme that it is z&axis" #he $iew plane

normal $ector is the $ector 3 anPrincipal -anishing Point is

P

1 ?a3 0 ?b3

/ ?c3@ d3 n

%

"here#ore,-?-3>@-8 A@-B

1?-3t@l

0?-8t@m

/?-t @nP(l,m,n)B?"-?n3$3@n8$8@n$

=here a3,b3,c3 are coordinates o the centre o projection

n3

,n8

,n

are components o $iew plane normal $ector

d3 is proportional to distance D rom the $iew plane to the .entre o


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#wo Principal -anishing PointPerspecti$e Projection

“#his Perspecti$e projection occrswhen the projection plane intersectsexactly two o the principal axes"!

Corizon ine

VP1 VP2


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.ontineEEE""

>n this case where the projection plane

intersects the x and y axes, orexample, the

normal $ector satisfes the relationship"B?F or

n(?F, and so the principal $anishingpoints are

1 $a1% &1 n1

Y 1 $'1!1 $c1

P1%2 $a1 Y 2$'1% &1 n2!2 $c1

P2

%

#h P i i l - i hi P i t


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#hree Principal -anishing PointPerspecti$e Projection

• #his Perspecti$e projection occrs when

the projection plane intersects all thethree principal axes x, y and z& axes"P1

P3

P2

1 $a1% &1 n1

Y 1 $'1!1 $c1

P1

2 $a1 Y 2 $'1% &1 n2!2 $c1

P2

3 $a1 Y 3 $'1!3 $c1% &1 n3

P3


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Parallel Projection

Basic Principles% &

“ #he parallel projection sed bydraters and engineers to createwor:ing drawings o an object whichpreser$es its scale and shape" #hecomplete representation o these

details oten re'ires two or more$iews (projections) o the object ontodiGerent $iew planes"


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.ontineEEE"

• >n parallelprojection, imagepoints are ond as

the intersection othe $iew plane witha projector drawnrom the objectpoint and ha$ing afxed direction"

P(x,y,z)

P2

P’(x’,y’,z’)

P2’

!

Y

Direction o -

Projection

-iew Plane


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athematical Description

* Parallel projection is determined byprescribing a direction o projection$ector - and a $iew plane" #he $iew

plane is determined by its viewreference point +o and view plane

normal " the object point P is located

in world co&ordinates at (x,y,z)" #heproblem is to determine the image point co-ordinates P’(x’,y’,z’)"


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#ypes o Parallel Projection

*" 9rthographic Projection

7" 9bli'e Projection


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A*rthographicPro+ection

“Projections are characterized by the act that

the direction o projection is perpendiclar to

the $iewing plane" #hey are sed to prodce

theront, side and top $iews o an object"!

xa-ple% & 5ngineering < architectraldrawings

employ it"

.ategories2 # pes o


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.ategories2 #ypes o9rthographic

Projections3" ./lti0iew Pro+ections% &“=hen the direction

o projection is parallel to any o theprincipal axis, this prodces ront,top and side $iews o an object"!

xa-ple% &mechanical drawingsemploy it"


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.ontineEEEE

8" Axono-etric Pro+ections% &

“#hese projections are those in which

the direciton o projection is not parallel to any o

the three principal axis"!o-e co--on s/'categories o# Axono-etric

Pro+ections are% &

3" >sometric

8" Di&metric" #ri&metric


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.ontineEEEEE"

3" >sometric Projection% he direction oprojection

ma:es e'al angles with all the three principa

axes"8" Di&etric Projection% & #he direction o

projection ma:es e'al angles with exactlytwo o the principal axes"

" #ri&etric% & #he direction o projection ma:esne'al angles with the three principal axes"


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7" 9bli'e Projection

“Projection obtained by projecting pointsalong

parallel lines that are not perpendiclar to

$iewing plane i"e" at any angle oconsideration

is called obli'e parallel projections"!

*“on&orthographic parallel projections are

.alled obli'e parallel projections"!


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4BCA"5*6 *7A**."6C

P*8C"6* A% &

1 Ca0alier% &

“ #he direction o projections is chosenso that there is no ore&shortening o

lines perpendiclar to xy&plane"!

8" Ca'inet% &

“#he direction o projections is chosen

so that lines perpendiclar to xy&planeare ore&shortening by hal theirlengths"!


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.abinet Projection

y

x

C’

9’

’

:’

5’A’ B’

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Projection- parallel and perspective

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