Graphics Reference
In-Depth Information
Figure 14.37.
Offset curve generated by two-axis NC
machine.
Figure 14.38.
NC cutter over- and undercuts.
is a counter-clockwise oriented
unit
normal vector to this curve.
Definition.
The
offset curve
p
d
(u) to the planar curve p(u), which is a distance |d|
from p(u) for some real number d is defined by
d
()
=
()
+
()
p
u
p u
dn u
(14.28)
Since
d
ยข
()
=
()
+
(
()
)
pu pu
1 k
ud
,
(14.29)
where k(u) is the curvature function of p(u), we see that the tangent vector to p
d
(u)
is parallel to the tangent vector to p(u). One can also show that the curvature func-
tion for p
d
(u), k
d
(u), is defined by
()
k
k
u
ud
()
=
k
d
u
.
()
1
+
Its derivative with respect to arc-length parameterization is given by
d
ds
k
1
d
ds
k
d
d
=
.
(14.30)
3
(
)
1
+
k
d
Note that even if the original curve p(u) has a nice smooth shape, p
d
(u) may not
be. In Figure 14.34, the offset curves
C
1
and
C
2
are as nice as the original ellipse. On
