1

PROJECTIONS

1. Parallel Projectionsa) Orthographic Projectionsb) Axonometric Projections

2. Perspective Transformations and Projections

AML710 CAD LECTURE 8

PROJECTIONS� Affine, Rigid-body/Euclidian Vs Perspective� Both affine and perspective transformations are 3-

dimensional� They are transformations from one 3-D space to

another� Viewing 3D transformations (results) on a 2-

Dimensional surface(screen) and requires projections from 3-Space to 2-Space.

� This is known as plane geometric projection

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PROJECTIONS� Projections are a necessary part of Graphics

Pipeline

PROJECTIONRasterizationDisplay

ClippingOrthographic/

PerspectiveVisual RealismGeometrical Model

Viewing Transformations

Rendering/Shading

ModelingTransformations

Graphics Pipeline

PROJECTIONS - ClassificationPlane Geometric Projections

Parallel Perspective

Orthographic Axonometric Oblique

Trimetric

Dimetric

Isometric

Cavelier Cabinet

Single Pt Two Pt Three Pt

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PROJECTIONS – Parallel Vs Perspective

y

z

xInfinityProjectors

screen

Imag

e

Object

y

z

xCOP

Projectorsscreen

Imag

e

Object

Parallel Projection

Perspective Projection

�Generalized 4 x 4 transformation matrix in homogeneous coordinates

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Translations l, m, n along x, y, and z axisLinear transformations – local scaling, shear,

rotation reflectionPerspective transformationsOverall scaling

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�Orthographic projection matrices

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Orthographic Views

Y=0 (xz)On +ve y axisTopZ=0 (xy)On -ve z axisRearX=0 (yz)On -ve x axisLeft SideY=0 (xz)On -ve y axisBottom

X=0 (yz)On +ve x axisRight SideZ=0 (xy)On +ve z axisFrontProj. PlaneC.O.ProjectionView

Ortho graphic views

y

z

x

Top

Y=0

z

x

Right

X=0z

y

y

Front

Z=0y

InfinityProjectors

5

�Example – Auxiliary View

�Consider the position vector [X]�Direction cosines are

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Concatenated matrix

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βα

and

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AXONOMETRIC PROJECTIONS� The limitations of orthographic projections are

overcome� An axonometric projection is obtained by

manipulating the object, using rotation and translations such that at least 3 adjoining faces are shown. The result is then projected from COP at infinity onto one of the coordinate planes,usually on z=0

� Features� Unless the plane is parallel to the POP, an axonometric

projection does not show its true shape� Parallel lines are equally foreshortened and the relative

lengths of parallel lines remain constant

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TRIMETRIC PROJECTIONSArbitrary rotations in arbitrary order about any or all

of the coordinate axes, followed by parallel projections on z=0 plane. The ratios of lengths are obtained as:

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yyyxyf

xyxxxf

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�DIMETRIC & ISOMETRIC PROJECTIONSJust as in the case of trimetric projections, similar

transformations + projections cause dimetric and isometric projections with following conditions:

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DimetricsamearefffofAny

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7

Example: Trimetric projectionsConsider the following cube rotated by φ=30°about y

axis and θ=45°about x-axis followed by a parallel projection onto the z=0 plane. The position vectors for the cube with one corner removed are

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Example (contd.): Trimetric projectionsThe concatenated matrix is :

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Calculation of angles Let us consider the axonometric projection of unit vectors

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UTX

θφφ 2sin2sin2cos2'2'2 +=+= xyxxf x

lengthtruelengthprojected

factorningforeshorte =

θ2cos2'2'2 =+= yyyxf y

θφφ 2sin2cos2sin2'2'2 +=+= zyzxf z

Calculation of angles For dimetric projections fx = fy(say) then

θθφφ 2cos2sin2sin2cos22 =+�= yx ff

0)2sin1(4sin22sin2 2 =−−− zfθθθ

θθφ2sin1

2sin2sin−

=

The second equation is obtained in terms of fz and solving for theta

)2/(sin)2/(sin 211zfzfandzf −±=±= −− φθ

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From the above two equations, solving for theta

Substituting this in the above eqn., we obtain

Calculation of angles For Isometric projections fx = fy =fy =f then

θθφφ 2cos2sin2sin2cos22 =+�= yx ff

�26.3531sin3

1sin 2 ±=�±=�= θθθ

θθφ

θθφ

2sin1

2sin212sinand2sin1

2sin2sin−

−=−

=

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sin 21

31

31

2 ±==−

= φφ

8165.0cosfactorningForeshorte 322 === φf

Calculation of angle that the projected x-axis makes with the horizontal in isometric case

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factorningforeshorte =

°=±=== 45assincos

sinsintan *

*

φθφ

θφαx

x

xy

The angle between the projected x-axis and horizontal is given by

°±=±=∴ − 30)26.35sin(tan 1α

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�Perspective Transformations