This document discusses 3D transformations and projections. It describes two main projection methods: parallel projection and perspective projection. Parallel projection preserves proportions but does not provide a realistic 3D representation. Perspective projection maps 3D points along converging lines to a vanishing point, resulting in foreshortening effects where objects appear smaller the farther they are from the viewing plane. The document outlines different types of parallel and perspective projections.

1. 3-D Transformations

2. 3-D Projections
We can project the 3-D objects onto the 2-D
plane. So Projection can be defined as a
mapping of point P onto its image P’ in the
projection plane or view plane.
There are two basic projection methods:
 Parallel projection
 Perspective projection

3. Parallel Projection
 Coordinate positions are transformed to the
view plane along parallel lines. The image
points are found as the intersection of the
view plane with the projector.
View Plane
P2
P2’
P1 P1’

4.  Parallel Projection preserves relative
proportions of objects.
 Accurate views of the various sides of an
object are obtained with a parallel
projection, but this does not give us a
realistic representation of the appearance of
a 3-D object.
 We can specify a parallel projection with a
projection vector that defines the direction
for the projection lines.

5. Types of Parallel Projections:
(i) Orthographic Projection
(ii) Oblique projection
y P2
P1 Oblique
Projection
Orthographic
Projection
x
P2’
P2’
z P1’ P1’

6.  Orthographic parallel projection: When
the projection is perpendicular to the view
plane. And parallel to any of the principal
axis this produces the front, top and side
views. See next slide….

7.  Types of Orthographic projections:
(i) Axonometric projection: that display
more than one face of an object. Most
common axonometric is Isometric
projection.
Isometric projection is generated by aligning
the projection plane so that it intersects
each coordinate axis in which the object is
defined at the same distance from the
origin.The direction of projection makes
equal angles with all the principal axis.

9.  Oblique projection: If the direction of
projection is not perpendicular to the
projection plane.
 Types of Oblique Projection are:
(i) Cavalier- the direction of projection is
chosen so that there is no foreshortening
of lines perpendicular to the xy plane.
(ii) Cabinet- the direction of projection is
chosen so that lines perpendicular to the
xy planes are foreshortened by half their
lengths.

10. Perspective Projection
Points on the body of an object is 3-D are
transformed to the viewing plane along
lines that converge to a point called
vanishing point(center of projection).
C
Center
Of projection
(Vanishing
Point

11. So the distance of a line from the projection
plane determines its size on the projection
plane, i.e. the farther the line is from the
projection plane, the smaller its image on the
projection plane.
Characteristics:
(i) Vanishing Point: The lines that are
parallel to the viewing plane appear to
converge at a point called Vanishing point.

12. (ii) Perspective Foreshortening : Objects
that are farther from the viewing plane are
projected smaller in size than the objects
that are nearer to viewing plane.
(iii) View confusion : When we project
objects which are behind the center of
projection appears to be projected upside
down & backward onto the viewing plane.