IV Projection
• A projection transformation moves from three
dimensions to two dimensions
• Projections occur based on the viewpoint and
the viewing direction
• Projections move objects onto a projection
plane
• Projections are classified based on the
direction of projection, the projection plane
normal, the view direction, and the viewpoint
• Two primary classifications are parallel and
perspective
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Parallel Projection
• All projectors run parallel and in the viewing
direction
• Projecting onto the z = 0 plane along the z axis
results in all z coordinates being set to zero
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Orthographic Projection
• The simplest of the parallel projections, which is
commonly used for engineering drawings
• They accurately show the correct or true size
and shape of a single plane face of an object
• The matrix for projection
onto the x=0 plane
• The matrix for projection
onto the y=0 plane
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• The matrix for projection
onto the z=0 plane
• The centers of projection are at infinity
• A single orthographic projection does not
provide sufficient information to visually and
practically reconstruct the shape of an object
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Axonometric Projection
• An axonometric projection is constructed by
manipulating the object, using rotations and
translations, such that at least three adjacent
faces are shown
• The center of projection is at infinity
• Unless a face is parallel to the plane of
projection, an axonometric projection does not
show its true shape
• The foreshortening factor is the ratio of the
projected length of a line to its true length
• There are three axonometric projections of
interest: trimetric, diametric, and isometric
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• An isometric projection is a special case of a
diametric projection, and a diametric projection
is a special case of a trimetric projection
• Foreshortening factors along the projected
principal axes are:
• A trimetric projection is formed by arbitrary
rotations, in arbitrary order, about any or all of
the coordinate axes, followed by parallel
projection onto the z=0 plane
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• A dimetric projection is a trimetric projection with
two of the three foreshortening factors equal; the
third is arbitrary
• For example, a dimetric projection is constructed
by a rotation about the y-axis through an angle 
followed by rotation about the x-axis through an
angle  and projection from a center of projection
at infinity onto the z=0 plane:
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Let
2
sin  
2
fz
2
2
sin  
2
fz
2  fz
2
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• Isometric projection: All three foreshortening
factors are equal
sin    1 / 3
   35 . 26 
sin    1 / 2
   45 
• The foreshortening factor: f  2 / 3  0 . 8165
• The angle that the projected x-axis makes with
the horizontal is
  tan
1
(  sin  )   30 
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Oblique Projection
• An oblique projection is formed by parallel
projectors from a center of projection at infinity
that intersect the plane of projection at an
oblique angle
• Only faces of the
object parallel to
the plane of
projection are
shown at their
true size and
shape
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• Cavalier projection: The angle between the
oblique projectors and the plane of projection
is 45 . The foreshortening factors for all three
principal directions are equal to 1
• Cabinet projection: The foreshortening factor for
edges perpendicular to the plane of projection is
one-half. The angle between the projectors and
the plane of projection is cot  1 (1 / 2 )  63 . 43 
• The angle between the oblique projectors and
the plane of projection is   cot  1 ( f )
• The transformation for
an oblique projector is
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Perspective Projection
• The viewpoint is the center of projection for a
perspective projection
• In perspective transformations parallel lines
converge, object size is reduced with increasing
distance from the center of projection
• All of these
effects aid the
depth perception
of the human
visual system,
but the shape of
the object is not
preserved
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• Any of the first three elements of the fourth
column of the general 4  4 homogeneous
coordinate transformation matrix is nonzero
• The single-point perspective transformation with
centers of projection and vanishing points on the
z-axis:
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r  
1
zc
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• The center of projection is
• Perspective factor: r
[0
0
1/ r
1]
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•
0
0
1
1

0

0
0

0
0
0
1
0
0
1
0
0
• Vanishing point :
0

0
  0
r

1
x

0
y

r
1
z


1  0
0
1/ r
1
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• The single-point perspective transformation with
centers of projection and vanishing points on
the x-axis:
• The center of projection is [  1 / p 0 0 1]
• The vanishing point is [1 / p 0 0 1]
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• The single-point perspective transformation with
centers of projection and vanishing points on
the y-axis:
• The center of projection is [ 0  1 / q 0 1]
• The vanishing point is [ 0 1 / q 0 1]
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• Two-point perspective transformation: Two terms
in the first three rows of the fourth column of the
transformation
matrix are nonzero.
44
• Two centers of projection:
[1 / p
[0
• Two vanishing points: [1 / p
[0
0
0
1]
1/ q
0
1]
0
1]
1/ q
0
0
1]
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• Three-point perspective transformation: Three
terms in the first three rows of the fourth column
of the 4  4 transformation matrix are nonzero.
• Three centers of projection:
[0
1/ q
0
1]
[1 / p
0
0
1]
[0
1/ r
1]
• Three vanishing points: [1 / p
[0
1/ q
0
1] [ 0
0
0
0
0
1]
1/ r
1]
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• Consider simple translation of the object
followed by a single-point perspective projection
from a center of projection at z  z c onto the z=0
plane
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• Combine rotation and translation about a single
principal axis to obtain an adequate 3D
representation
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• A three-point perspective transformation can be
obtained by rotating about two or more of the
principal axes and then performing a single-point
perspective transformation
• For example, a rotation about the y-axis followed
by a rotation about the x-axis and a perspective
projection onto the z=0 plane from a center of
projection at z=zc
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Vanishing Point
• Principal vanishing points: Points on the
horizontal reference line at which lines originally
parallel to the untransformed principal axes
converge, when a perspective view of an object
is created by using a horizontal reference line,
normally at eye level.
• In general, different sets of parallel lines have
different principal vanishing points
• Trace points: For planes of an object which are
tilted relative to the untransformed principal axes,
the vanishing points fall above or below the
horizontal reference line. These are often called
trace points.
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• The methods for determining the vanishing points:
– The intersection point of a pair of transformed projected
parallel lines
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– The transformation of the points at infinity on the
principal axes
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• The transformation of the points at infinity for
skew planes can be used to find trace point
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Reconstruction of 3D Images
• The general perspective transformation is:
• The transformation projected onto the z=0 plane is
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• If the location of several points which appear in
the perspective projection are known in object
space and in the perspective projection, then it is
possible to determine the transformation
elements
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• Project I: Please show the effect of transformation,
such as translation, rotation, reflection, parallel
projection and perspective projection on a cube
with one corner cut off.
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