3D PROJECTION
OUTLINE
• Classification of different types of projections
• Parallel projection
• Perspective projection
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PROJECTION
• Geometric projection:
• Projectors are straight
lines.
• Planar projection:
• Projection surface is a
plane.
Projection
Geometric
Planar
Non-planar
Non-
geometric
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PLANAR GEOMETRIC PROJECTION
• Parallel:
• Different projectors
parallel to each other.
• Preserve parallelism.
• Perspective:
• Different projectors
converge to the center of
projection.
• Does not preserve
parallelism.
Planar Geometric
Projection
Parallel
Orthographic
Oblique
Perspective
One-point
Two-point
Three-point
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PARALLEL PROJECTION
• Orthographic:
• Projectors are
perpendicular to the
projection plane.
• Oblique:
• Projectors are NOT
perpendicular to the
projection plane.
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Projectionsurface
Projectors
ORTHOGRAPHIC PROJECTION
• Elevation:
• Projectors are parallel to
a principle axis.
• Axonometric:
• Projectors are NOT
parallel to any principle
axis.
Orthographic
Elevation
Top
Front
Side
Axonometric
Isometric
Dimetric
Trimetric
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ELEVATIONS
• Used for:
• Engineering drawings.
• Architecture drawings.
• Advantage:
• Preserve distance and angle.
• All views are at same scale.
• Disadvantage:
• Hard to understand the 3D shape even all three elevations are given.
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X
Y
O
Z
Top
Front
MATRIX REPRESENTATIONS
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AXONOMETRIC PROJECTION
• Isometric:
• Angles between the
projections of all three
principal axes are equal
(120º).
• Dimetric:
• Angles between the
projections of two of the
principal axes are equal.
• Trimetric:
• All three angles are different.
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Isometric
Dimetric
ISOMETRIC PROJECTION
• Used for:
• Patent office records;
• Furniture design;
• Video games (SimCity).
• Advantage:
• Illustrates 3D nature of
object.
• Preserve distance along
principal axes.
• Disadvantage:
• Do not preserve angle.
• Lack of foreshortening.
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OBLIQUE PROJECTION
Oblique
Cavalier
Cabinet
Other
shear
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Projection Plane
Projectors
α
φ
Projectors
CAVALIER & CABINET PROJECTIONS
12
α=45°
Cavalier
Cabinet
α=tan-12≈63°
1
½
1
1
1
½
MATRIX REPRESENTATION
• Cavalier:
• u = x + z*cos(φ)
• v = y + z*sin(φ)
• Cabinet:
• u = x + z*cos(φ)/2
• v = y + z*sin(φ)/2
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PERSPECTIVE PROJECTION
• Foreshortening effect:
• Objects further from the
center of projection
appear smaller.
• Vanishing point:
• Projections of parallel
lines that are not parallel
to the projection plane
converge to a point.
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Projectionsurface
Projectors
Center ofProjection
ONE-POINT & TWO-POINT PROJECTIONS
• One-point projection:
• The projection plane is
parallel to two of the
principle axes.
• Two-point projection:
• The projection plane is
parallel to only one of
the principle axes.
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RENAISSANCE ART
• Brunelleschi invented
method of determining
perspective projections in
early 1400’s.
• Dutch artist, Johannes
Vermeer, may used camera
obscura to create
perspective effect.
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Lady Standing at the Virginals
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Albrecht Dürer, Daraughsman Drawing a Recumbent Woman (1525) Woodcut illusion from 'The Teaching of Measurements
MATRIX REPRESENTATION I
• Configuration:
• The center of projection
is placed at (0,0)
• The function for
projection plane is z=f:
• For 3D point (x,y,z,1):
• w = z / f;
• u = x / w = x * f / z
• v = y / w = y * f / z
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Projectionplane
Z
X
u(x,z)
Center ofProjection
MATRIX REPRESENTATION II
• Configuration:
• The center of projection is
placed at (0,-f);
• The function for projection
plane is z=0:
• For 3D point (x,y,z,1):
• w = z / f + 1;
• u = x / w
• = x * f / (z + f)
• v = y / w
• = y * f / (z + f)
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Projectionplane
Z
X
u(x,z)
Center ofProjection