Lissajous Curve
Lissajous curves are the family of curves described by the parametric equations
sometimes also written in the form
They are sometimes known as Bowditch curves after Nathaniel Bowditch, who studied them in 1815. They were studied in more detail (independently) by Jules-Antoine Lissajous in 1857 (MacTutor Archive). Lissajous curves have applications in physics, astronomy, and other sciences. The curves close iff omega_x/omega_y is rational.
Lissajous curves are a special case of the harmonograph with damping constants beta_1=beta_2=0.
Special cases are summarized in the following table, and include the line, circle, ellipse, and section of a parabola.
It follows that omega=2, delta=pi/2 gives a parabola from the fact that this gives the parametric equations (acos(2t),sint)=(a(1-2sin^2t),sint)=(a-2asin^2t,sint), which is simply a horizontally offset form of the parametric equation of the parabola (u^2/(4a),u).
See also
Harmonograph, Simple Harmonic MotionExplore with Wolfram|Alpha
More things to try:
References
Cundy, H. and Rollett, A. "Lissajous's Figures." §5.5.3 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 242-244, 1989.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 70-71, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178-179 and 181-183, 1972.MacTutor History of Mathematics Archive. "Lissajous Curves." https://mathshistory.st-andrews.ac.uk/Curves/Lissajous/.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, p. 142, 1991.Referenced on Wolfram|Alpha
Lissajous CurveCite this as:
Weisstein, Eric W. "Lissajous Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LissajousCurve.html