TRANSFORMATIONS AND PROJECTIONS 49computer. Though recent developments in computer graphics facilitate better appreciation of an
object in three-dimensions, projective geometry or engineering graphics still plays a vital role in
visual communication. Engineering graphics is developed using the theory of projections that allow
representing three-dimensional objects on two-dimensional planes.
Projections can be primarily classified as perspective and parallel. Projective geometry operates
using: (a) location of the eye in three-dimensional space with respect to the object, also called the
view point and (b) location of the plane of projection or the image or picture plane, in relation to the
object. A line of sight is an imaginary ray of light between the view point and the object. In perspective
projection, all lines of sight commence from a single point. The view point is at a finite distance from
the object, and the lines of sight connecting the view point to the boundaries of the object are not
parallel. On the contrary, in parallel projection, the lines of sight are parallel, or the view point is
stationed at infinity in relation to the object. The plane of projection is imaginary upon which the rays
along the lines of sight impinge and create points corresponding to the boundaries or the interior
features of the object. Joining such points on the plane systematically creates a trace or image of the
object. This plane may either be the computer screen (in modern day practice) or a piece of paper (in
a conventional set-up).
Perspective projections are closest to what a human eye visualizes. However, they are difficult to
construct, and it is also difficult to obtain realistic dimensions of the object for its creation or
manufacture. Parallel projections are less realistic, but are easier to draw. It is easier to communicate
the actual dimensions and manufacturing details through parallel projections. Orthographic projection
is a parallel projection technique in which the plane of projection is positioned perpendicular to the
lines of sight. Orthographic projections can either be axonometric or multi-view. Axonometric projections
provide a three-dimensional view of the object and can be classified into isometric, dimetric or tri-
metric. Multi-view orthographic projections provide two-dimensional views of the object, and many
such views are required to obtain its comprehensive three-dimensional appreciation. This method is
more popular in engineering as multi-view projections give true dimensions without much further
calculations. They provide an accurate description for manufacturing and construction. A technician
can easily be trained to read multi-view orthographic drawings without requiring of him to have an
artistic acumen. The rest of the chapter discusses the theoretical aspects of generic perspective and
parallel projections, with emphasis on orthographic projections. Aerial perspective is beyond the
scope of this chapter. Classification of projections is provided in Figure 2.20.
2.7.1 Geometry of Perspective Viewing
In perspective viewing, the image plane is placed between the view point and the object. Although,
this is not a restriction, for the object between the view point and image plane, a reversed image is
formed. The eye should neither lie on the object nor on the image plane. The image plane need not,
in general, be perpendicular to the object plane. For the image plane as planar, we obtain a linear
perspective projection though the image plane may as well be spherical, cylindrical or a part of any
generic curved surface.
Consider a point P (x,y,z) on the object (Figure 2.21) and E as the observer’s eye located at
(0, 0, –w) on the z-axis. Let the image plane be the x-y plane and the line segment EP intersect the
image plane at P (x,y, 0). Let PB = x and PD = y. For P′ as the foot of the perpendicular from
P to the x-y plane, P′C = y and P′A = x. For similar triangles POE and P*P′P
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PPOP
PPEP
′′PP