Parallel Projections - Transformations and Projections in Computer Graphics - page 59

Example :Given s z =1 / 2, we calculate θ and φ :
θ =sin − 1

=sin − 1 (

0 . 5
√ 2

20 . 7 ◦ , ± ± 0 . 35355) = ±

φ =sin − 1

=sin − 1 (

0 . 5

22 . 2 ◦ .

√ 2

± ± 0 . 378) = ±

− 0 . 5 2

The two rotations are illustrated in Figure 2.7.

y

x

(a) (b) (c)

Figure 2.7: Rotations for Dimetric Projection.

Exercise 2.2: Repeat the example for s z =0 . 625.

Exercise 2.3: Calculate θ and φ for s x = s z (equal shrink factors in the x and z
directions).

The condition for an isometric projection (Figure 2.6c) is s x = s y = s z .Wealready
know that s x = s y results in Equation (2.5). Similarly, it is easy to see that s y = s z
results in cos 2 θ =sin 2 φ +cos 2 φ sin 2 θ , which can be written

2sin 2 θ

sin 2 φ = 1

−

. (2.7)

sin 2 θ

1 −

Equations (2.5) and (2.7) result in sin 2 θ =1 2sin 2 θ or sin 2 θ =1 / 3, yielding θ =

−

35 . 26 ◦ . The rotation angle φ can now be calculated from Equation (2.5):

±

1 / 3

sin 2 φ = 45 ◦ .

1 / 3 =1 / 2 ,

yielding φ = ±

1 −

The shrink factors can be calculated from, for example, s y =cos 2 θ = 2 / 3 0 . 8165.

We conclude that the isometric projection is the most useful but also the most
restrictive of the three axonometric projections. Given a diagram with the isometric
projection of an object, we can measure distances on the diagram and divide them by

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