Graphics Reference
In-Depth Information
Figure 12.9.
Sweep surfaces: a cutter path and square rotated along a spiral.
Definition. The parametric surface p(s,t) defined by equation (12.15b) is called the
sweep surface obtained by sweeping f(t) along the framed curve g(s).
12.5.1 Example. The path of a cutter as shown in Figure 12.9(a) can easily be
described as two sweep surfaces defined by equations like equation (12.15b), where
each function f(t) parameterizes a vertical segment.
Equation (12.15b) can be generalized by letting a one-parameter family of
affine maps T s act on the curve f(t) as we sweep it along g(s). Define function
g i (s,t) by
(
()
) =
(
() () ()
)
T f t
g st g st g st
,,
,,
,
s
1
2
3
and the new parametric surface p(s,t) by
() =
() +
() () +
() () +
() ()
pst
,
g
s
g st
,
u
s
g st
,
u
s
g st
,
u
s
.
(12.16)
1
1
2
2
3
3
Definition. The parametric surface p(s,t) defined by equation (12.16) is called a
screw sweep surface .
12.5.2
Example.
Let g(s) be the spiral
() = (
)
g s
3
cos , sin ,
s
3
s s
.
Using the Frenet frame (T(s),N(s),B(s)) for this curve we let
() =
() =-
(
)
u
sNs
cos ,
s
-
sin ,
s
0
,
1
() = (
) (
() =
)
u
sTs
110 3
sin ,
s
3
cos ,
s
1
,
2
() = (
)
() =
(
)
u
sBs
110
sin ,
s
-
cos ,
s
3
.
3
Let f : [0,4] Æ R 3 parameterize the unit square in the x-z plane as follows:
Search WWH ::




Custom Search