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3D Concepts
UNIT 3
3-D Coordinate Spaces
• Remember what we mean by a 3-D coordinate
space
y axis
y
P
z
x
z axis
x axis
Right-Hand
Reference System
Translations In 3-D
• To translate a point in three dimensions by tx,
ty and tz simply calculate the new points as
follows:
• x’ = x + tx
y’ = y + ty
z’ = z + tz
(x, y, z)
(x’, y’, z’)
Translated Position
Scaling In 3-D
• To scale a point in three dimensions by sx, sy
and sz simply calculate the new points as
follows:
• x’ = sx*x
y’ = sy*y
z’ = sz*z
(x’, y’, z’)
(x, y, z)
Scaled Position
Rotations In 3-D
• The equations for the three kinds of rotations
in 3-D are as follows:
x’ = x·cosθ - y·sinθ
y’ = x·sinθ + y·cosθ
z’ = z
x’ = x
y’ = y·cosθ - z·sinθ
z’ = y·sinθ + z·cosθ
x’ = z·sinθ + x·cosθ
y’ = y
z’ = z·cosθ - x·sinθ
Homogeneous Coordinates In 3-D
• Similar to the 2-D situation we can use
homogeneous coordinates for 3-D
transformations - 4 coordinate y axis
column vector
y
• All transformations can
then be represented
as matrices
P
P(x, y, z) =
x
 
y
 
z
 
1 
z
x
z axis
x axis
3D Transformation Matrices
Translation by
tx, ty, tz
1

0

0

0
0
0
cos 
 sin 
sin 
cos 
0
0
1

0

0

 tx
0

0

0

1
0
0
1
0
0
1
ty
tz
 sx

0

0

0
0

0

0

1
 cos 

0

  sin 

 0
0
sin 
1
0
0
cos 
0
0
0
0
sy
0
0
sz
0
0
0

0

0

1
0

0 Scaling by

0  sx, sy, sz

1
 cos 

sin 

 0

 0
 sin 
0
cos 
0
0
1
0
0
0

0

0

1
Projections
• Our 3-D scenes are all specified in 3-D world
coordinates
• To display these we need to generate a 2-D
image - project objects onto a picture plane
Picture Plane
Objects in
World Space
• So how do we figure out these projections?
Converting From 3D To 2D
• Projection is just one part of the process of
converting from 3D world coordinates to a 2D
image
3-D world
coordinate
output
primitives
Clip against
view volume
Project onto
projection
plane
Transform to
2-D device
coordinates
2-D device
coordinates
3D Viewing
• In 2D viewing we have 2D window & 2D viewport &
objects in the world coordinates.
• The 3D viewing has an added dimension which makes
it complex as even though objects are 3D the display
devices are only 2D.
• The mismatch between 3D objects & 2D displays is
compensated by introducing projections. The
projection transforms 3D objects into a 2D projection
plane.
• View plane: It is nothing but the film plane in a camera
which is positioned & oriented for a particular shot of
the scene.
• World coordinates positions in the scene are
transformed to viewing coordinates, then viewing
coordinates are projected onto the view plane.
• View reference point: This point is the center
of our viewing coordinate system.
• The production of a 2D image of higher
dimensional object refers to graphical
projection.
• A projection can be defined as a mapping of
any point P[x,y,z] to its image P`[x`,y`,z`] onto
the view plane, called as projection plane.
• Parallel & perspective projections are the two
Types of Projections
• There are two broad classes of projection:
– Parallel: Typically used for architectural and
engineering drawings.
– Perspective: Realistic looking and used in
computer graphics.
Parallel Projection
Perspective Projection
Parallel Projections
• Some examples of parallel projections
Orthographic Projection
Isometric Projection
Perspective Projections
• Perspective projections are much more realistic
than parallel projections
Geometric
projection
Parallel
Orthogr
aphic
To
p
Fro
nt
Perspective
Oblique
Ax
on
om
etr
ic
Other
Twopoint
Side
elev
atio
n
Cabinet
Isometric
Onepoint
Cavalier
Other
Threepoint
• Parallel projection:
– If the direction of projection is perpendicular to the
projection plane, it is an orthographic projection.
– If the direction of projection is not perpendicular to
the projection plane is called as oblique projection.
– A multi-view projection displays a single face of a 3D
object.
– Axonometric projections allow the user to place the
view-plane normal in any direction such that 3
adjacent faces of a cube like object are visible.
– Dimetric projections differ from isometric
projections in the direction of the view-plane
normal.
– Trimetric projections allow the viewer the most
freedom in selecting the components of n.
• Perspective projection:
– It is a type of projection where 3D objects are not
projected along parallel lines, but along lines
emerging from a single point.
– A vanishing point is a point in a perspective drawing
to which parallel lines appear to converge.
– One-point perspective exists when a painting plate is
parallel to two axes of a rectilinear scene.
– Two point perspective
Assignment
Orthographic
Wireframe
Elevation
Orthographic
Wireframe
Plan
Orthographic
Wireframe
End-Elevation
Perspective
View