Based on the observation that many surfaces of revolution can be treated as a canal surface, i.e., the envelope surface of a one-parameter family of spheres, the analytical expressions of the envelopes of the swept volumes generated by the commonly used rotary cutters undergoing general spatial motions are derived by using the envelope theory of sphere congruence. For the toroidal cutter, two methods for determining the effective patch of the envelope surface are proposed. With the present model, it is shown that the swept surfaces of a torus and a cylinder can be easily constructed without complicated calculations, and that the minimum distance (between the swept surface and a simple surface) and the signed distance (between the swept surface and a point in space) can be easily computed without constructing the swept surface itself. An example of global tool path optimization for flank milling of ruled surface with a conical tool, which requires to approximate the tool envelope surface to the point cloud on the design surface, is given to confirm the validity of the proposed approach.

1.
Chiou
,
C. J.
, and
Lee
,
Y. S.
, 2005, “
Optimal Tool Orientation for Five-Axis Too-End Machining by Swept Envelope Approach
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
127
(
4
), pp.
810
818
.
2.
Chiou
,
C. J.
, 2004, “
Accurate Tool Position for Five-Axis Ruled Surface Machining by Swept Envelope Approach
,”
Comput.-Aided Des.
0010-4485,
36
(
10
), pp.
967
974
.
3.
Lartigue
,
C.
,
Duc
,
E.
, and
Affouard
,
A.
, 2003, “
Tool Path Deformation in 5-Axis Flank Milling Using Envelope Surface
,”
Comput.-Aided Des.
0010-4485,
35
(
4
), pp.
375
382
.
4.
Wang
,
W. P.
, and
Wang
,
K. K.
, 1986, “
Geometric Modeling for Swept Volume of Moving Solids
,”
IEEE Comput. Graphics Appl.
0272-1716,
6
(
12
), pp.
8
17
.
5.
Abdel-Malek
,
K.
, and
Yeh
,
H. J.
, 1997, “
Geometric Representation of the Swept Volume Using Jacobian Rank-Deficiency Conditions
,”
Comput.-Aided Des.
0010-4485,
29
(
6
), pp.
457
468
.
6.
Abdel-Malek
,
K.
,
Seaman
,
W.
, and
Yeh
,
H. J.
, 2001, “
NC Verification of Up to 5 Axis Machining Processes Using Manifold Stratification
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
123
(
1
), pp.
99
109
.
7.
Blackmore
,
D.
,
Leu
,
M. C.
, and
Wang
,
L. P.
, 1997, “
The Sweep-Envelope Differential Equation Algorithm and Its Application to NC-Machining Verification
,”
Comput.-Aided Des.
0010-4485,
29
(
9
), pp.
629
637
.
8.
Leu
,
M. C.
,
Wang
,
L. P.
, and
Blackmore
,
D.
, 1997, “
A Verification Program for 5-Axis NC Machining With General APT Tools
,”
CIRP Ann.
0007-8506,
46
(
1
), pp.
419
424
.
9.
Leu
,
M. C.
,
Wang
,
L. P.
, and
Blackmore
,
D.
, 1998, “
Simulation of NC Machining With Cutter Deflection by Modeling Deformed Swept Volume
,”
CIRP Ann.
0007-8506,
47
(
1
), pp.
441
446
.
10.
Hu
,
Z. J.
, and
Ling
,
Z. K.
, 1996, “
Swept Volumes Generated by the Natural Quadric Surfaces
,”
Comput. Graphics
0097-8493,
20
(
2
), pp.
263
274
.
11.
Jüttler
,
B.
, and
Wagner
,
M. G.
, 1996, “
Computer-Aided Design With Spatial Rational B-Spline Motions
,”
ASME J. Mech. Des.
0161-8458,
118
(
2
), pp.
193
201
.
12.
Xia
,
J.
, and
Ge
,
Q. J.
, 2001, “
On the Exact Representation of the Boundary Surfaces of the Swept Volume of a Cylinder Undergoing Rational Bezier and B-Spline Motions
,”
ASME J. Mech. Des.
0161-8458,
123
(
2
), pp.
261
265
.
13.
Chung
,
Y. C.
,
Park
,
J. W.
,
Shin
,
H.
, and
Choi
,
B. K.
, 1998, “
Modeling the Surface Swept by a Generalized Cutter for NC-Verification
,”
Comput.-Aided Des.
0010-4485,
30
(
8
), pp.
587
593
.
14.
Roth
,
D.
,
Bedi
,
S.
,
Ismail
,
F.
, and
Mann
,
S.
, 2001, “
Surface Swept by a Toroidal Cutter During 5-Axis Machining
,”
Comput.-Aided Des.
0010-4485,
33
(
1
), pp.
57
63
.
15.
Mann
,
S.
, and
Bedi
,
S.
, 2002, “
Generalization of the Imprint Method to General Surfaces of Revolution for NC Machining
,”
Comput.-Aided Des.
0010-4485,
34
(
5
), pp.
373
378
.
16.
Chiou
,
C. J.
, and
Lee
,
Y. S.
, 2002, “
Swept Surface Determination for Five-Axis Numerical Control Machining
,”
Int. J. Mach. Tools Manuf.
0890-6955,
42
(
14
), pp.
1497
1507
.
17.
Weinert
,
K.
,
Du
,
S. J.
,
Damm
,
P.
, and
Stautner
,
M.
, 2004, “
Swept Volume Generation for the Simulation of Machining Processes
,”
Int. J. Mach. Tools Manuf.
0890-6955,
44
(
6
), pp.
617
628
.
18.
Du
,
S. J.
,
Surmann
,
T.
,
Webber
,
O.
, and
Weinert
,
K.
, 2005, “
Formulating Swept Profiles for Five-Axis Tool Motions
,”
Int. J. Mach. Tools Manuf.
0890-6955,
45
(
7–8
), pp.
849
861
.
19.
Zhu
,
L. M.
,
Zheng
,
G.
, and
Ding
,
H.
, 2009, “
Formulating the Swept Envelope of Rotary Cutter Undergoing General Spatial Motion for Multi-Axis NC Machining
,”
Int. J. Mach. Tools Manuf.
0890-6955,
49
(
2
), pp.
199
202
.
20.
Aras
,
E.
, 2009, “
Generating Cutter Swept Envelopes in Five-Axis Milling by Two-Parameter Families of Spheres
,”
Comput.-Aided Des.
0010-4485,
41
(
2
), pp.
95
105
.
21.
Bianchi
,
L.
, 1927,
Lezioni di Geometria Differenziale
,
3rd ed.
,
Nicola Zanichelli Editore
,
Bologna
.
22.
Yoshizawa
,
S.
,
Belyaev
,
A.
, and
Seidel
,
H. P.
, 2007, “
Skeleton-Based Variational Mesh Deformations
,”
Comput. Graph. Forum
1067-7055,
26
(
3
), pp.
255
264
.
23.
Xu
,
Z. Q.
,
Feng
,
R. Z.
, and
Sun
,
J. G.
, 2006, “
Analytic and Algebraic Properties of Canal Surfaces
,”
J. Comput. Appl. Math.
0377-0427,
195
(
1-2
), pp.
220
228
.
24.
Pigel
,
L.
, and
Tiller
,
W.
, 1995,
The NURBS Book
,
Springer
,
Berlin
.
25.
Lee
,
K.
,
Seong
,
J. K.
,
Kim
,
K. J.
, and
Hong
,
S. J.
, 2007, “
Minimum Distance Between Two Sphere-Swept Surface
,”
Comput.-Aided Des.
0010-4485,
39
(
6
), pp.
452
459
.
26.
Sarkar
,
B.
, and
Menq
,
C. H.
, 1991, “
Parameter Optimization in Approximating Curves and Surfaces to Measurement Data
,”
Comput. Aided Geom. Des.
0167-8396,
8
(
4
), pp.
267
290
.
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