This paper describes a mechanism design methodology that draws plane curves which have finite Fourier series parameterizations, known as trigonometric curves. We present three ways to use the coefficients of this parameterization to construct a mechanical system that draws the curve. One uses Scotch yoke mechanisms for each of the terms in the coordinate trigonometric functions, which are then added using a belt or cable drive. The second approach uses two-coupled serial chains obtained from the coordinate trigonometric functions. The third approach combines the coordinate trigonometric functions to define a single-coupled serial chain that draws the plane curve. This work is a version of Kempe's universality theorem that demonstrates that every plane trigonometric curve has a linkage which draws the curve. Several examples illustrate the method including the use of boundary points and the discrete Fourier transform to define the trigonometric curve.

References

References
1.
Coros
,
S.
,
Thomaszewski
,
B.
,
Noris
,
G.
,
Sueda
,
S.
,
Forberg
,
M.
,
Sumner
,
R. W.
,
Matusik
,
W.
, and
Bickel
,
B.
,
2013
, “
Computational Design of Mechanical Characters
,”
ACM Trans. Graphics
,
32
(
4
), p.
83
.
2.
Thomaszewski
,
B.
,
Coros
,
S.
,
Gauge
,
D.
,
Megaro
,
V.
,
Grinspun
,
E.
, and
Gross
,
M.
,
2014
, “
Computational Design of Linkage-Based Characters
,”
ACM Trans. Graphics
,
33
(
4
), p.
64
.
3.
Nolle
,
H.
,
1974
, “
Linkage Coupler Curve Synthesis: A Historical Review—II: Developments After 1875
,”
Mech. Mach. Theory
,
9
(
3–4
), pp.
325
348
.
4.
Nolle
,
H.
,
1974
, “
Linkage Coupler Curve Synthesis: A Historical Review—I: Developments Up To 1875
,”
Mech. Mach. Theory
,
9
(
2
), pp.
147
168
.
5.
Koetsier
,
T.
,
1983
, “
A Contribution to the History of Kinematics—I: Watt's Straight-Line Linkages and the Early French Contributions to the Theory of the Planar 4-Bar Coupler Curve
,”
Mech. Mach. Theory
,
18
(
1
), pp.
37
42
.
6.
Koetsier
,
T.
,
1983
, “
A Contribution to the History of Kinematics—II: The Work of English Mathematicians on Linkages During the Period 1869-78
,”
Mech. Mach. Theory
,
18
(
1
), pp.
43
48
.
7.
Kempe
,
A. B.
,
1876
, “
On a General Method of Describing Plane Curves of the nth Degree by Linkwork
,”
Proc. London Math. Soc.
,
VII
(
102
), pp.
213
216
.
8.
Kempe
,
A. B.
,
1877
,
How to Draw a Straight Line
,
Macmillan
,
London
.
9.
Jordan
,
D.
, and
Steiner
,
M.
,
1999
, “
Configuration Spaces of Mechanical Linkages
,”
Discrete Comput. Geom.
,
22
(
2
), pp.
297
315
.
10.
Kapovich
,
M.
, and
Millson
,
J. J.
,
2002
, “
Universality Theorems for Configuration Spaces of Planar Linkages
,”
Topology
,
41
(
6
), pp.
1051
1107
.
11.
Kobel
,
A.
,
2008
, “
Automated Generation of Kempe Linkages for Algebraic Curves in a Dynamic Geometry System
,”
BCS thesis
, Saarland University, Saarbrucken, Germany.
12.
Saxena
,
A.
,
2011
, “
Kempe's Linkages and the Universality Theorem
,”
Resonance
,
16
(
220
), pp.
220
237
.
13.
Liu
,
Y.
, and
McCarthy
,
J. M.
,
2017
, “
Synthesis of a Linkage to Draw a Plane Algebraic Curve
,”
Mech. Mach. Theory
,
111
, pp.
10
20
.
14.
Hong
,
H.
, and
Schicho
,
J.
,
1998
, “
Algorithms for Trigonometric Curves (Simplification, Implicitization, Parameterization)
,”
J. Symbolic Comput.
,
26
(
3
), pp.
279
300
.
15.
Artobolevskii
,
I. I.
,
1964
,
Mechanisms for the Generation of Plane Curves
,
Pergamon Press
,
London
.
16.
Miller
,
D. C.
,
1916
, “
A 32-Element Harmonic Synthesizer
,”
J. Franklin Inst.
,
181
(
1
), pp.
51
81
.
17.
Nie
,
X.
, and
Krovi
,
V.
,
2005
, “
Fourier Methods for Kinematic Synthesis of Coupled Serial Chain Mechanisms
,”
ASME J. Mech. Des.
,
127
(
2
), pp.
232
241
.
18.
Oppenheim
,
A. V.
, and
Schafer
,
R. W.
,
2010
,
Discrete-Time Signal Processing
,
Pearson Higher Education
,
Upper Saddle River, NJ
.
19.
Kendig
,
K.
,
2011
,
A Guide to Plane Algebraic Curves
,
MAA
,
Washington, DC
.
20.
Shikin
,
E. V.
,
1995
,
Handbook and Atlas of Curves
,
CRC Press
,
Boca Raton, FL
.
21.
Fay
,
T. H.
,
1989
, “
The Butterfly Curve
,”
Am. Math. Mon.
,
96
(
5
), pp.
442
443
.
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