Graphics Reference
In-Depth Information
Figure 14.41.
Offset curves for multi-axes milling machines.
2
2
(
()
)
+-
()
(
)
2
sxu
-
tyu
-=
d
0.
(14.35)
Condition (14.34) then translates into the constraint that the tangent vector is orthog-
onal to the radius vector, namely,
¢
()
(
()
-
()
)
=
pu
•
s xut yu
-
,
0
(14.36)
As a final application involving offset curves, consider the problem of milling free-
form surfaces. This sort of operation requires minimally a three-axis milling machine
that can move in three orthogonal directions. See Figure 14.41(a). Better yet is a five-
axis milling machine that has an additional two degrees of freedom to allow rotations
specified by two angles. See Figure 14.41(b). In any case, to carry out the milling, one
can first mill along the boundary curves of the patch. Then along an offset curve in
the interior of the patch that is an offset of the boundary. We can continue this way
until the whole surface is milled. This process involves defining offset curves for space
curves. This time we do not have well-defined normals to offset along since there is a
whole normal plane at each point of the curve. However, if we look at circles of radius
d in these normal planes, what we want is that the offset curve intersects that circle
at some point. It is not hard to write down the appropriate equations.
Next, let p(u) be a space curve and (T(u),N(u),B(u)) its Frenet frame. The princi-
pal normal N(u) and binormal B(u) form an orthonormal basis for the normal plane
to the curve at p(u).
Definition.
An
offset curve
p
d
(u) to the space curve p(u) at a distance d > 0 from p(u)
is defined by an equation of the form
d
()
=
()
+
(
()
+
()
)
pu pu d
cos
q
Nu
sin
q
Bu
,
(14.37)
where q is a function of u in general.
Approaches to computing offset curves for a surface patch parameterized by a
function p(u,v) are discussed in [HosL93]. One practical problem with the formula-
tion above is that the principal normal and binormal are not easy to compute.
This concludes our discussion of offset curves. For additional facts see [FarN90a],
[FarN90b], or [MaeP93]. Offsets of clothoidal splines are discussed in [MeeW90]. For
an overview of a different approach, where one tries to approximate the offset rather
than represent it exactly, see [ElLK97]. The motivation is that working with approxi-
