Computer Aided Engineering Design

TRANSFORMATIONS AND PROJECTIONS 57

If all three foreshortening factors are equal, we get an isometric projection. For shy = shz = shx and
using the above equations

sin =

sin

1 – sin

, also sin =

1 – 2 sin

1 – sin

sin =^1

3

(^2) = 35.26 = 45
2
2
2
2
φ 2
ψ
ψ
φ
ψ
ψ
⇒±⇒±°⇒±°ψψ φ
Thus, the foreshortening factor for an isometric projection is given by shy = shz = shx = sh =
1 – sin^2 ψ =^2 = 0.8165
3

. For an isometric projection of a machine part, we can measure the

dimensions on the figure and divide it by 0.8165 to obtain the actual dimensions of the object. A
rotation of ± 45 ° about the y-axis and ±35.26° about the x-axis gives a tilted object with respect to the
x-y plane. The object is placed such that its principal edges or axes make equal angles with the x-y
plane. The edges are thus foreshortened in equal proportions to 81.65%. Thus, for a cube, the edges
will appear to be at 120° (or 60°) with respect to each other in the projection. Projecting the unit
vectori = [1 0 0 0] along x*-axis attached to the tilted cube on to the plane of projection gives

cos sin

sin sin cos – cos sin

00 00

00 01

φφ

φψ ψ φψ

00

0

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

[1 0 0 0]T = [cos φ sin φ sin ψ 0 0]T

This is a vector on the plane (x-y) of projection and passing through the origin O. The angle α
betweenOx* and the projected line on the plane of projection is given by

tan =

sin sin

cos

=

sin 45 sin

cos 45

α = sin = tan ( sin 35.26 ) = 30–1

φψ

φ

ψ

ψα

°

°

±⇒ ± °±°

In drawing an isometric scale, first a base line L is made and then a line l at 45° to the base line.
The true scale is drawn on l. Another line m is drawn at 30° to L and the true scale is projected
froml to m. This is called the isometric scale. Isometric projections have the following general
characteristics: (a) parallel edges on the object remain parallel in the isometric projection, (b)
vertical edges of the object remain vertical in the projection and (c) all horizontal lines appear at
30 ° with the horizontal.

Example 2.12. A prismatic machine block is composed of 10 planar surfaces with vertices having the
following homogenous coordinates:

P1 = [0 0 0 1; 6 0 0 1; 6 3 0 1; 0 3 0 1; 0 0 0 1]
P2 = [0 0 0 1; 0 0 3 1; 2 0 3 1; 2 0 2 1; 6 0 2 1; 6 0 0 1; 0 0 0 1]
P3 = [0 0 3 1; 2 0 3 1; 2 1 5 1; 0 1 5 1; 0 0 3 1]
P4 = [0 1 5 1; 2 1 5 1; 2 2 5 1; 0 2 5 1; 0 1 5 1]
P5 = [2 2 5 1; 2 3 3 1; 0 3 3 1; 0 2 5 1; 2 2 5 1
P6 = [2 3 3 1; 0 3 3 1; 0 3 0 1; 6 3 0 1; 6 3 2 1; 2 3 2 1; 2 3 3 1]
P7 = [2 0 2 1; 2 3 2 1; 6 3 2 1; 6 0 2 1; 2 0 2 1]
P8 = [0 0 0 1; 0 0 3 1; 0 1 5 1; 0 2 5 1; 0 3 3 1; 0 3 01; 0 0 0 1]
P9 = [2 0 2 1; 2 0 3 1; 2 1 5 1; 2 2 5 1; 2 3 3 1; 2 3 2 1; 2 0 2 1]
P10 = [6 0 0 1; 6 0 2 1; 6 3 2 1; 6 3 0 1; 6 0 0 1]