Graphics Programs Reference
In-Depth Information
such that all its sides seem to have the same length. This type of axonometric projection
is called isometric .
Matrix T of Equation (2.2) can be used to calculate the special rotations that
produce a dimetric projection. Consider the product of a unit vector in the x direction
and T :
cos φ
sin φ sin θ
0
=(cos φ, sin φ sin θ, 0) .
(1 , 0 , 0)
0
cos θ
0
(2.3)
sin φ
cos φ sin θ
0
This product shows that any vector in the x direction shrinks, after being rotated by
matrix T ,byafactor s x given by Equation (2.4). The same equation also produces the
shrink factors s y and s z of any vector in the y and z directions.
s x = cos 2 φ +sin 2 φ sin 2 θ,
s z = sin 2 φ +cos 2 φ sin 2 θ.
s y = cos 2 θ,
(2.4)
If we want a dimetric projection where equal-size segments in the x and y directions
will have equal sizes after the projection, we set s x = s y or, equivalently,
cos 2 φ +sin 2 φ sin 2 θ =cos 2 θ,
which produces the relation
sin 2 θ
1 sin 2 θ .
sin 2 φ =
(2.5)
Equation (2.5) together with the expression for s z yields
s z =sin 2 φ +cos 2 φ sin 2 θ =sin 2 φ +(1
sin 2 φ )sin 2 θ
=sin 2 φ (1 sin 2 θ )+sin 2 θ
sin 2 θ
sin 2 θ )+sin 2 θ,
=
sin 2 θ (1
1
or 2 sin 4 θ
(2 + s z )sin 2 θ + s z = 0, a quadratic equation in sin 2 θ whose solutions are
sin 2 θ = s z / 2andsin 2 θ = 1. The second solution cannot be used in Equation (2.5) and
has to be discarded. The first solution produces
φ =sin 1
.
θ =sin 1
s z
2
s z
2
±
and
±
(2.6)
s z
1 , 1], the arg u ment of sin 1 must
Since the sine function has values in the range [
be in this range. T he expr ession s z / 2isinthisrangewhen
2
+ 2, and
s z
the expression s z / 2
+1. Since s z is a shrinking
factor, it is nonnegative, which implies that it must be in the interval [0 , 1]. Also, since
Equation (2.6) contains a
s z
is in this range when
1
s z
±
,anyvalueof s z produces four solutions.
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