As
in
two-dimensional viewing, the projection operations can take place
be
fore the view-volume clipping or after clipping. All obpcts within the view vol-
ume map to the interior of the specified projection window. The last step is to
transform the window contents to a two-dimensional viewport, which
specifies
the location of the display on the output device.
Clipping in two dimensions is generally performed against an upright
rec-
tangle; that
is,
the dip window is aligned with the
x
and
y
axes.
This
greatly
sirn-
plifies the clipping calculations, because .each window boundary is defined by
one coordinate value. For example, the intersections of all
lines
crossing the left
boundary of the window have an
x
coordinate equal to the left boundary.
View-volume clipping boundaries are planes whose orientations depend on
the
type
of projection, the propchon window, and the position of the projection
reference point. Since the front and back clipping planes are parallel to the view
plane, each has a constant z-coordinate value. The
z
coordinate of the inters-
tions of lines with these planes
is
simply the
z
coordinate of the corresponding
plane. But the other four sides of the view volume can have arbitrary spatial ori-
entations. To find the intersection of a line with one of the view volume bound-
aries means that we must obtain the equation for the plane containing that
boundary polygon.
This
process
is
simplified
if
we convert the view volume
be
fore clipping to a rectangular parallelepiped. In other words, we first perform the
projection transformation, which converts coordinate values in the view volume
to orthographic parallel coordinates, then we carry out the clipping calculations.
Clipping against a regular parallelepiped
is
much simpler because each sur-
face is now perpendicular to one of the coordinate axes. As seen in Fig.
12111,
the
top and bottom of the view volume are now planes of constant
y,
the sides
are
planes of constant
x,
and the front and back are planes of constant
z.
A
line cut-
ting
through the top plane of the parallelepiped, for example, has an intersection
point whose y-coordinate value
is
that of the top plane.
In the case of an orthographic parallel projection, the view volume
is
al-
ready a rectangular parallelepiped.
As
we have seen in Section
12-3,
obliquepro-
jechon view volumes are converted to a rectangular parallelepiped
by
the shear-
ing operation, and perspective view volumes are converted, in general, with a
combination shear-scale transformation.
v
Figure
12-41
An
obpct
intersecting
a
rectangular
parallelepiped view volume.

Chapter
1 2
Normalized
View
Volumes
Three-Dimensional Mewing
Figure
12-42
shows the expanded
PHIGS
transformation pipeline. At the first
step, a scene is constructed by transforming obpct descriptions from modeling
coordinates to world coordinates. Next, a view mapping convert: the world de-
scriptions to viewing coordinates. At the projection stage, the viewing coordj-
nates are transformed to projection coordinates, which effectively converts the
view volume into a rectangular parallelepiped. Then, the parallelepiped is

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