Computer Aided Engineering Design

60 COMPUTER AIDED ENGINEERING DESIGN

2.9 Oblique Projections

In axonometric projections, the parallel rays or projectors are perpendicular to the plane of projection.
If these projectors are inclined at an angle to the plane of projection, the image obtained is the oblique
projection. It is sometimes found useful to show the three-dimensional details of the object through
oblique projections. Faces parallel to the plane of projection are not foreshortened and the angles
between the edges of the parallel faces are also preserved. However, faces not parallel to the plane
of projection, get distorted.
Two types of oblique projections popular in engineering drawing are: (a) cavalier and (b) cabinet.
The geometry of oblique projections can be explained by assuming three orthogonal axes and unit
vectorsi (1, 0, 0), j (0, 1, 0), k (0, 0, 1). Let the tip of k be designated by the point A and the projection
plane be the x-y plane. Consider a set of parallel rays at an angle θ to the x-y plane (from behind the
z-axis as shown in Figure 2.30). A parallel ray through A intersects the x-y plane at B (bx,by, 0). Let
angleBOX be ψ. Consider any point C (0, 0, z) on the z-axis such that a parallel ray through C
intersects the x-y plane at D (dx,dy, 0). Here OB = f is the shrink factor. From triangles AOB and
COD, we can get the following relationships:

bx = f cos ψ, by = f sin ψ, f = cot θ, dx = fz cos ψ, dy = fz sin ψ (2.38)

Figure 2.29 Three orthographic views in third angle

Top view

Front view Right side view

B

x

D

y

O θ

A

C

z

ψ

Figure 2.30 Geometry for oblique projections