AML710 CAD
LECTURE 8
PROJECTIONS
1. Parallel Projections
a) Orthographic Projections
b) Axonometric Projections
2. Perspective Transformations and
Projections
PROJECTIONS
Affine, Rigid-body/Euclidian Vs Perspective
Both affine and perspective transformations are 3dimensional
They are transformations from one 3-D space to
another
Viewing 3D transformations (results) on a 2Dimensional surface(screen) 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
Modeling
Transformations
Rendering/
Shading
Geometrical Model Visual Realism
Display
Viewing
Transformations
Orthographic/
Perspective
Rasterization
Clipping
PROJECTION
Graphics Pipeline
PROJECTIONS - Classification
Plane Geometric Projections
Parallel
Orthographic
Perspective
Axonometric
Trimetric
Single Pt
Oblique
Cavelier
Two Pt
y
y
Cabinet
Projectors
Dimetric
Isometric
COP
Infinity
Projectors
t
ecct
bbjje
O
O
Three Pt
e
agge
ma
IIm
screen
screen
xx
z
z
2
Generalized 4 x 4 transformation matrix in
homogeneous coordinates
[T ] =
a
d
g
l
b
c
e
f
i
j
m
n
p
q
r
s
Translations l, m, n along x, y, and z axis
Linear transformations – local scaling, shear,
rotation reflection
Perspective transformations
Overall scaling
Orthographic projection matrices
1 0 0 0
[Tz ] =
0 1 0 0
0 0 0 0
1 0 0 0
; [ Ty ] =
0 0 0 1
0 0 0 0
0 0 1 0
0 0 0 0
; [Tx ] =
0 0 0 1
0 1 0 0
0 0 1 0
0 0 0 1
Orthographic Views
View
Front
Right Side
Top
Rear
Left Side
Bottom
C.O.Projection
On +ve z axis
On +ve x axis
On +ve y axis
On -ve z axis
On -ve x axis
On -ve y axis
Proj. Plane
Z=0 (xy)
X=0 (yz)
Y=0 (xz)
Z=0 (xy)
X=0 (yz)
Y=0 (xz)
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Ortho graphic views
x
Top
y
Y=0
z
Projectors
y
x
Infinity
Right
z
y
X=0
z
Front
Z=0
y
Example – Auxiliary View
Consider the position vector [X]
Direction cosines are c x c y c z =
[
] [
y
∴α = +450
1
and β = +35.260
Concatenated matrix
1 0 0 0
[T1 ] = [ Pz ][T ] =
2
0 1 0 0
0 0 0 0
0 0 0 1
6
0
2
0
6
1
3
1
3
3
]
x
z
2
2
6
6
2
0
6
2
6
2
6
2
0
6
2
0
0
0
1
=
6
0
2
6
2
0
0
0
0
6
2
6
2
0
0
6
0
0
0
1
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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 atleast 3 adjoint 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
TRIMETRIC PROJECTIONS
Arbitrary 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:
xx'
x 'y
x z'
0 1 0
y x'
[T ][U ] = [T ]
=
0 0 1
0
1 1 1
1
y 'y
y z'
0
1
1
1
1 0 0
fx =
x 'x2 + y 'x2
f y = x 'y2 + y 'y2
fz =
foreshortening factor =
projected length
true length
x 'z2 + y 'z2
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DIMETRIC & ISOMETRIC PROJECTIONS
Just as in the case of trimetric projections, similar
transformations + projections cause dimetric and
isometric projections with following conditions:
fx ≠ f y ≠ fz
Any
Trimetric
2 of f x f y f z are same
fx = f y = fz
Dimetric
Isometric
Example: Trimetric projections
Consider 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
0 1
[ X ]=
1
0 0 0.5
0.5 0 0 1 1 0
1
1 0 0 1 1
1
1
1 1
1
1
1 0 0 0 0 0.5
1 1
1
1
1 1 1 1 1
[T ] = [ Pz ][ R x ][ R y ] =
0
1
1
0 1 0 0
0 cosϑ
−sin ϑ 0
0 0 0 0
0 sin ϑ
cosϑ
0
− sin φ
0 0 0 1
0
0
1
0
0
0
0
cos φ
1 0 0 0
0
0 sin φ
0
1
0
0
0 cosφ
0
0
1
0
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Example (contd.): Trimetric projections
The concatenated matrix is :
[T ] = [ Pz ][ R x ][ R y ] ==
0.5
[ X ′] =
cos φ
0
sin φ
0
sin φ sin θ
cosθ
− cos φ sin θ
0
0
0
0
0
0
0
0
1
1.366
1.366 0.933
0.5
3
=
2
0
2
4
0
0 0.866 0.866
1
0
2
0
2
0
0
0
2
6
4
0
0
0
1
0
1.116
− 0.612 − 0.259 0.095 0.272 0.095 0 0.354 1.061 0.707 0.754
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
Calculation of angles Let us consider the
axonometric projection of unit vectors
[ X ′] = [T ][U ] ==
cos φ
0
sin φ
0 1 0 0
sin φ sin θ
cos θ
− cos φ sin θ
0 0 1 0
0
0
0
0 0 0 1
0
0
0
1 1 1 1
foreshortening factor =
f x2 = x '2 + y '2 = cos 2 φ + sin 2 φ sin 2 θ
x
x
projected length
true length
f y2 = x '2 + y '2 = cos 2 θ
y
y
f z2 = x '2 + y '2 = sin 2 φ + cos 2 φ sin 2 θ
z
z
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Calculation of angles For dimetric projections fx = fy
(say) then
f = f
x
y
cos 2 φ + sin 2 φ sin 2 θ = cos 2 θ
sin 2 θ
2
sin φ =
1 − sin 2 θ
The second equation is obtained in terms of fz and
solving for theta
2 sin 2 θ − 2 sin 4 θ − (1 − sin 2 θ ) f 2 = 0
z
θ = sin −1 (± f / 2 )
and
z
φ = sin −1 (± f / 2 − f z2 )
z
Calculation of angles For Isometric projections
fx = fy =fz =f then
f = f
cos 2 φ + sin 2 φ sin 2 θ = cos 2 θ
x
y
sin 2 φ =
sin 2 θ
1 − sin 2 θ
and
sin 2 φ =
1 − 2 sin 2 θ
1 − sin 2 θ
From the above two equations, solving for theta
sin 2 θ = 1
sin θ = ± 1
θ = ±35.26
3
3
Substituting this in the above eqn., we obtain
sin 2 φ =
1
= 12
3
1−
1
φ = ±45
3
Foreshortening factor f = cos 2 θ =
2
3
= 0.8165
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Calculation of angle that the projected x-axis makes
with the horizontal in the isometric case
[U *] = [U ][T ] = [1 0 0 1]
= [cos φ sin φ sin θ
cos φ
sin φ sin θ
0 0
0
cos θ
0 0
sin φ
− cos φ sin θ
0 0
0
0
0 1
0 1]
foreshortening factor =
projected length
true length
The angle between the projected x-axis and horizontal is given by
y *x sin φ sin θ
tan α = * =
= ± sin θ as φ = 45°
xx
cos φ
∴α = tan −1 (± sin 35.26) = ±30°
Perspective Transformations
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