Computer Aided Engineering Design

TRANSFORMATIONS AND PROJECTIONS 55

We may observe that for projection on the x-y plane, the entire third row of Prxy is 0. Similarly,
for projections on the y-z and z-x planes, the entire first and second rows of Pryz and Przx, respectively,
are zero. We can obtain six views for six sides of the object enclosed inside an imaginary cube
using these transformations. Usually, two or three views are adequate to show all features of the
object. However, for some objects with curved surfaces, projections on auxiliary planes may be
required. An auxiliary plane is not parallel to any coordinate plane and a unit normal for it is first
obtained. The object is then manoeuvered till the normal to the auxiliary plane is coincident with
one of the coordinate axes. The respective projection transformation in Eq. (2.36) is applied, and
then concatenating the inverse transformations from the left places the object back to its original
location.
Two schemas in wide use in orthographic projections are: (a) the first angle and (b) the third angle
projections. The object is enclosed in an imaginary cube and parallel projections are taken from the
object to planes of the cube. In the first angle projection, the projections pass through the object to
intersect the plane behind while in the third angle schema, projections reflect back onto the plane in
front. Planes with projections are then unfolded to show the required views in two dimensions. The
first angle projections of an object are shown in Figure 2.25 as an example.

Figure 2.25 An object’s orthographic projections in first angle

Top view

Back-side view Right view Front view Left view

Object Bottom view

2.8.1 Axonometric Projections

In an axonometric projection, the object is rotated about any axis and translated, if desired, till the
view reveals more than two faces of the object. Projections are then made with the eye at infinity
(parallel rays of projection) and one of the coordinate planes as the plane of projection. Any other
plane of projection may also be chosen so long as the rays are perpendicular to it. Axonometric
projection contains more geometric information about the object than an orthographic projection in