Graphics Reference
In-Depth Information
Figure 13.5.
The fat line for a Bézier curve.
2
2
ax
++=
by
c
0
,
a
+=
b
1
,
(13.8)
and define
(
)
=++
dxy
,
ax by c
.
(13.9)
It is easy to see that d(x,y) is just the signed distance from a point (x,y) to
L
because
n
= (a,b) is a unit normal vector for
L
and d(x,y) = ((x,y) -
p
0
) • n. Let d
i
= d(x
i
,y
i
) and
set
{
}
d
=
min
d
,
d
,...,
d
,
min
01
n
{
}
d
=
max
d
,
d
,...,
d
.
max
01
n
The convex hull property of Bézier curves implies that the curve is contained in the
fat line defined by the two lines
L
1
and
L
2
, which are parallel to
L
and a distance of
d
min
and d
max
from
L
, respectively. Actually, as the example in Figure 13.5 indicates,
we can do better. The curve is contained in a thinner fat line and the lines
L
1
and
L
2
can be moved closer to
L
. The thinnest fat line parallel to
L
is really determined by
the minimum and maximum of d(u), the signed distance from p(u) to
L
. The poly-
nomial d(u) can be explicitly computed in the quadratic and cubic curve case and one
can use the following improved values for d
min
and d
max
:
d
d
Ó
˛
Ó
˛
1
1
d
=
min
0
,
and
d
=
max
0
,
Quadratic Curve.
min
max
2
2
Cubic Curve.
Although precise values can be computed in this case also, the for-
mulas are complicated and the extra computational effort is not justified. Instead
the following approximations are suggested in [SedN90]:
If d
1
d
2
> 0, then let
3
4
3
4
{
}
{
}
d
=
min
0
,
d
,
d
and
d
=
max
0
,
d
,
d
.
min
12
max
12
