Parallel Projections - Transformations and Projections in Computer Graphics - page 57

in parallel. We assume that the projection plane is the xy plane, so the projection is
done by clearing the z coordinates of all the points or, equivalently, by multiplying each
point, after rotating it, by matrix T z of Equation (2.1). Assuming that we first rotate
the object φ degrees about the y axis and then θ degrees about the x axis, the combined
rotation/projection matrix is [see Equation (1.30)]

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

cos φ sin φ

01 0
sin φ

0 − 1 0 0 100

010
000

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

T = 0 θ sin θ

0 φ 0 − sin θ cos θ

⎛ ⎞

cos φ sin φ sin θ 0

⎝ ⎠ .

= 0 cos θ 0 (2.2)

sin φ − cos φ sin θ 0

To find how various dimensions are affected by these transformations, we start with the
vector (1 , 0 , 0). This is a unit vector in the direction of the x axis. Multiplying it by T

gives another vector, which we denote by ( x 1 ,x 2 , 0). Its magnitude is s x = x 1 + x 2
and since the original vector had magnitude 1, the quantity s x expresses the ratio of
magnitudes or the factor by which all dimensions in the x direction have shrunk after
the transformation/projection T . Similarly, selecting unit vectors (0 , 1 , 0) and (0 , 0 , 1)
in the y and z directions and multiplyin g them by T produc es vecto rs ( y 1 ,y 2 , 0) and

( z 1 ,z 2 , 0) and shrinking factors s y = y 1 + y 2 and s z = z 1 + z 2 in the y and z
directions, respectively.
Figure 2.6a shows a unit cube rotated such that its three sides, which used to be
parallel to the coordinate axes, seem to have different lengths. Such an axonometric
projection is called trimetric .

(a) (b) (c)

Figure 2.6: The Three Types of Axonometric Projections.

Figure 2.6b shows the same unit cube rotated such that two of its three sides
seem to have the same length, while the third side looks shorter. Such an axonometric
projection is called dimetric . Similarly, Figure 2.6c shows the same unit cube rotated