Lissajous curves are created by two simple harmonic motions — combinations of sine functions and/or cosine functions.
The curves are named after Jules-Antoine Lissajous, who studied them in 1857. They are sometimes called Bowditch curves after Nathaniel Bowditch, who discovered them (independently) in 1815 while studying a compound’s pendulum movement .
Lissajous Curve Equations
The parametric equations are combinations of sines and cosines:
- sin(nt + a), cos(kt + b) or
- sin(nt + a), sin(kt + b) or
- cos(nt + a), cos(kt + b) or
It’s common for the curves to be described by equations where a + b = 0. For example,
- x = cos(nθ), sin(k&theta)
You can play around with various values for n and k with this Desmos calculator page (which I used to create the top image).
Properties of the Lissajous Curve
The constants n and k determine the curve’s size.
One oscillation determines the x-coordinate and the other the y-coordinate.
When the ratio of the two oscillations’ frequencies is rational number (i.e., a/b) the curve is a closed curve. Different shapes are created with different ratios . When the frequencies are equal, the curve is a line segment (if the difference in phases is a multiple of π) or an ellipse.
There are two types of Lissajous curves :
- Type I curves are smooth curves.
- Type II have sharp ends.
Lissajous curves are Chebyshev polynomials , under certain conditions:
- n = 1,
- k = a natural number, and
Despite these straightforward properties, predicting the shape of the curve is challenging and may require some experimentation to achieve a certain shape .
 L.R. Ingersoll Physics Museum. Lissajous Curve. Retrieved February 28, 2022 from: https://www.physics.wisc.edu/ingersollmuseum/exhibits/waves/lissajous/
 MacTutor. Lissajous Curves. Retrieved February 28, 2022 from: https://mathshistory.st-andrews.ac.uk/Curves/Lissajous/
 Project: Lissajous Figures. Retrieved February 28, 2022 from: http://mathserver.neu.edu/~bridger/U170/Lissajous/Lissajous.htm
 Merino, J. (2003). Lissajous Figures and Chebyshev Polynomials. The College Mathematics Journal.
 Seggern, D. (1994). PHB practical handbook of curve design and generation. CRC Press.
Stephanie Glen. "Lissajous Curves (Bowditch Curve)" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/lissajous-curves/
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