Graphics Reference
In-Depth Information
Figure 4.16.
A parallel projection onto the
x-y plane.
Proof.
Exercise 4.9.1.
Passing to homogeneous coordinates, consider the projective transformation T
par
defined by the matrix
1
0
0
0
Ê
ˆ
Á
Á
Á
Á
˜
˜
˜
˜
0
1
0 0
10
M
par
=
(4.12)
v
v
v
v
1
3
2
3
-
-
Ë
¯
0
0
0
1
Our parallel projection onto the x-y plane is then nothing but the Cartesian version
of T
par
followed by the orthogonal projection (x,y,z) Æ (x,y,0). It follows that the matrix
M
par
plays the role of the matrix M
persp
in Section 4.5 (equation (4.9)) in that it reduces
a general projection problem into a simple orthogonal projection.
Notice that a parallel projection does not depend on the length of the vector
v
. In
fact, any multiple of
v
will define the same projection, as is easily seen from its equa-
tions. The parallel projection can also be considered the limiting case of a central pro-
jection where one places an eye at a position
v
= (v
1
,v
2
,v
3
) = (a¢d,b¢d,-d) and one lets
d go to infinity. This moves the eye off to infinity along a line through the origin with
direction vector
v
. The larger d gets, the more parallel are the rays from the eye to
the points of an object. The matrix M
eye
in equation (4.11) (with a = a¢d and b = b¢d)
approaches M
par
because 1/d goes to zero.
An even simpler case occurs when the vector
v
is orthogonal to the view plane.
Definition.
A parallel projection where the lines we are projecting along are orthog-
onal to the view plane is called an
orthographic
(or
orthogonal
)
projection
. If the lines
have a direction vector that is not orthogonal to the view plane, we call it an
oblique
(parallel) projection
. A view of the world obtained via an orthographic or oblique pro-
jection is called an
orthographic
or
oblique view
, respectively.
A single projection of an object is obviously not enough to describe its shape.
Definition.
An
axonometric projection
consists of a set of parallel projections that
shows at least three adjacent faces. A view of the world obtained via an axonometric
projection is called an
axonometric view
.
