Signal Processing - November 2017 - 163

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FIGURE 12. Mystery curves in 3-D. (a) The original 2-D mystery curve. (b) The 3-D mystery curve made from (a) by setting z (t) = 1 + cos (5t) . (c) The
3-D curve looks like a pineapple. (d) The 3-D curve looks like a pineapple from another point of view. This curve is fivefold symmetric so that we use
five different colors to indicate each fold. (e) Another 3-D mystery curve. This is a threefold symmetric curve, distinguished by the colors RGB, and the
rotation axis is not z-axis. (f) The same curve as (e) with a different angle to show symmetry.

What we have learned
In this article, we have learned some
properties and results of the mystery
curve. By introducing the polar form of
such a signal, which is related to a special case of AM-FM signal, we can now
easily design these beautiful curves and
the repeated patterns can be changed
flexibly. We have also learned how to
approximate the curve by taking finite
terms after low-pass filter. By the
Euler rotation theorem, the results we
achieved from 2-D cases can extend to
3-D cases directly.

Authors
Soo-Chang Pei (peisc@ntu.edu.tw)
received the B.S. degree from National
Taiwan University, Taipei, in 1970 and
the M.S. and Ph.D. degrees from the
University of California, Santa Barbara
in 1972 and 1975, respectively, all in
electrical engineering. Currently, he is
a professor in the Electrical Engineering Department at National Taiwan
University. His research interests include digital signal processing, image

processing, optical information processing, and laser holography. He is an
IEEE Life Fellow and a member of
Eta Keppa Nu and the Optical Society
of America.
Kuo-Wei Chang (muslimchang@
gmail.com) received the B.S. degree
from the Department of Electrical Engineering in 2004 and the M.S. degree from
the Graduate Institute of Communication
Engineering in 2006, both from National
Taiwan University, Taipei, where he is a
Ph.D. student in communication engineering. He is currently a researcher with
Telecommunication Laboratories, Chunghwa Telecom Co., Ltd. Some of his current research areas include digital signal
processing, music information retrieval,
and pattern recognition.

References

[1] F. Farris, Creating Symmetry: The Artful
Mathematics of Wallpaper Patterns. Princeton, NJ:
Princeton Univ. Press, 2015.
[2] B. Yorgey. (2015). Mystery curve, animated.
[Online]. Available: https://mathlesstraveled.com/
2015/06/05/mystery-curve-animated/
[3] K.-W. Chang. (2017). Mystery curve codes.
[Online]. Available: https://github.com/KuoWei
Chang/MysteryCurve

IEEE SIGNAL PROCESSING MAGAZINE

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November 2017

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[4] B. Yorgey. (2015). Mystery curve, animated.
[Online]. Available: https://mathlesstraveled
.com/2015/06/04/random-cyclic-curves-5/
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[10] R. Sharma, L. Vignolo, G. Schlotthauer, M.
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[11] C. T. Nguyen and J. P. Havlicek, "On the amplitude and phase computation of the AM-FMimage
model," in Proc. 2014 IEEE Int. Conf. Image
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[12] G. Biagetti, P. Crippa, A. Curzi, S. Orcioni, and
C. Turchetti, "Analysis of the EMG signal during
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[13] B. Palais, R. Palais, and S. Rodi, "A disorienting
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https://mathlesstraveled http://www.demonstrations.wolfram http://www.scipython.com/blog/the-mystery-curve/ https://www.mathlesstraveled.com/ https://www.github.com/KuoWei
Table of Contents for the Digital Edition of Signal Processing - November 2017
