Graphics Reference
In-Depth Information
A
A
P
P
e
r
B
B
(a)
(b)
Figure 11.6
(a) After accounting for errors and rounding to machine-representable numbers,
the computed intersection point of the segment
AB
and the plane
P
is unlikely to lie on either
line or plane. Instead, the point could lie in, say, any of the indicated positions. (b) By treating
the plane as having a thickness, defined by a radius
r
, the point is guaranteed to be on the
plane as long as
r
>
e
, where
e
is the maximum distance by which the intersection point can
be shown to deviate from
P
.
d
AB
A
d
P
P
B
e
e
(a)
(b)
Figure 11.7
(a) Let
AB
be a segment nearly perpendicular to a plane
P
. When
AB
is displaced
by a small distance
d
, the error distance
e
between the two intersection points is small. (b)
As the segment becomes more parallel to the plane, the error distance
e
grows larger for the
same displacement
d
.
plane, a small displacement
d
in the position of the segment will correspond to
a similar size displacement
e
of the intersection point (Figure 11.7a). However, as
the segment becomes increasingly parallel to the plane the same displacement of
the segment causes the error in the intersection point to grow disproportionately
(Figure 11.7b). Assuming there is a fixed upper bound for the length of the seg-
ment (which holds for most practical applications), having a thick plane means that
there is also a bound on how parallel to the plane the segment can become before
actually lying in the plane. Because no intersection computation is required once
the segment lies in the plane, the thick plane consequently enforces a limit on the
displacement error for the intersection point. Thick planes allow, for example, the clip-
ping of polygons to a plane to be performed robustly, as discussed in Sections 8.3.3
and 8.3.4.
