Signal Processing - November 2017 - 158

LECTURE NOTES
Soo-Chang Pei and Kuo-Wei Chang

The Mystery Curve: A Signal Processing Point of View

I

n the first chapter of a recently published book on artful mathematics
[1], a linear combination of harmonic
signals called mystery curves were introduced. Although their Fourier-based
analysis brings interesting results, this
lecture note provides a different and
important perspective, especially useful
for our signal processing community.
Based on polar coordinate systems and
low-pass filtering approaches, the patterns of the curves can be designed by
locally curve tracing instead of trial and
error, including not only two-dimensional (2-D) but also three-dimensional
(3-D) modeling. Concrete examples and
online MATLAB codes are provided
so that applications from art and logo
design to amplitude modulation (AM)frequency modulation (FM) signal
analysis are realizable.

Relevance
In [1], the first example given is a summation of three rotating circles n (t)
cos (6t) sin (14t)
n (t) = c cos (t) +
+
,
2
3
	
sin (6t) cos (14t)

sin (t) +
+
m,
2
3
(1)
which is illustrated in Figure 1. This
fivefold symmetric curve seems incomprehensible by the frequency components one, six, and 14.
Digital Object Identifier 10.1109/MSP.2017.2740457
Date of publication: 13 November 2017

158

However, it can be shown that the
symmetry comes from the difference of
frequency and can be easily proved by
Fourier series analysis. In this example,
if we put n on the complex plane

2
1

c (t) = x (t) + iy (t)
cos (6t) sin (14t)
= c cos (t) +
+
m
2
3
sin (6t) cos (14t)
+ i c sin (t) +
+
m

2
3
i6t
-14t
.
= e it + e + ie
2
3

1 - 6 =-5
6 - (-14) = 20
1 - (-14) = 15
and the greatest common divisor (gcd)
of all the aforementioned frequency differences is five, which is also the number
of folds. An equivalent statement in [1] is
that (1, 6, - 14) are all congruent to one
modulo five. In addition, if the frequencies are chosen randomly, as the code
shown in mysterycurve_rand_nofold.m
[3], we can observe that the curves in
Figure 2 are not always N-fold symmetry. More examples can be found in [4].
As a result, the relationship between the
frequencies and N -fold symmetry can
be deduced.
Although the author of [1] solved the
mystery and one could generate random
mystery curves by setting parameters,
it is still hard to synthesize a desired

|

-1
-2
-2

(2)

As we can see, the differences of frequency (1, 6, - 14) are

IEEE SIGNAL PROCESSING MAGAZINE

0

November 2017

|

-1

0

1

2

FIGURE 1. The mystery curve. An animated version can be found in [2].

N-fold pattern without trial and error. In
this article, we propose an another way
to generate the mystery curve. Instead
of finding a j and wj from
	

c (t) = / a j e iw j t, (3)
j

we use the polar coordinate system
	

c (t) = r (t) e i i (t) . (4)

After deriving the properties and constraints of r (t) and i (t), we can create a
pattern by locally tracing a curve on the
polar coordinate system. This is quite
useful for an art or commercial application such as logo design.
The beautiful N-fold symmetry
curve can be extended to 3-D cases
by similar definition in two dimensions. Today, 3-D printing is popular
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