56 COMPUTER AIDED ENGINEERING DESIGN
a single view. However, true dimensions are not shown as there is a foreshortening of the dimensions
depending upon the placement of the object. Three types of axonometric projections of interest are:
(a) trimetric, (b) dimetric and (c) isometric; the latter being more popular in use.
2.8.1.1 Trimetric Projection
Consider a cube placed with one corner at the origin and three of its orthogonal edges coincident with
the coordinate axes. The cube is rotated by an angle φ about the y-axis and ψ about the x-axis, and
its projection is taken on the x-y plane with the eye placed at infinity along the z-axis (parallel
projection-rays). Matrix M = PrxyRx(ψ)Ry(φ) provides the final transformation with
M=1000
0100
0000
000110 0 0
0 cos – sin
0 sin cos 0
00 0 1cos 0 sin 0
0100- sin 0 cos 0
0001
=cos sin
sin sin cos – cos sin⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥ψψ
ψψφφφφφφ
0 φψ ψ φψ00
00
00 00
00 01⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥(2.37)
Since the cube rests at a corner on the x-y plane, the projections of the sides are no longer of original
length as they are foreshortened. The foreshortening ratios shx,shy and shz can be determined as the
magnitudes of resultant vectors after transformation in Eq. (2.37) are applied to the three edges of the
cube, that is
cos sin
sin sin cos – cos sin
00 00
00 011001
0101
0011=cos sin sin 0 1
0 cos
sin – cos sin 0 1φφ
φψ ψ φψφφψ
ψ
φφψ00
0
01⎡⎣⎢
⎢
⎢
⎢⎤⎦⎥
⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥T Twhich gives the respective foreshortened ratios as
shxyz= cos^22 φφψ + (sin sin ) , sh = | cos |, ψsh = sin^2 φ φψ + (– cos sin )^2For a trimetric projection, all three foreshortening factors are unequal.
2.8.1.2 Dimetric Projection
In a dimetric projection, any two foreshortening factors are equal. Thus, for shy = shz
sin^2 φ + (– cos φ sin ψ)^2 = cos^2 ψ, also, shx^2 = cos^2 φ + (sin φsinψ)^2Adding together, we get1 + sin = cos + sin =
2
sin =
2cos =
2 –2222
2
2
ψψsh ψ ψ φ
sh sh sh
shxxx x
x⇒⇒±⇒±The result suggests that for a value of a given foreshortening factor, there are four possible combinations
ofφ and ψ and thus four possible diametric projections.
2.8.1.3 Isometric Projection
In engineering drawings, especially in mechanical engineering, isometric projections are used extensively.