Lissajous Curve

In mathematics, a Lissajous curve ( /ˈlɪsəʒuː/), also known as Lissajous figure or Bowditch curve ( /ˈbaʊdɪtʃ/), is the graph of a system of parametric equations

which describe complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous (a French name pronounced ) in 1857.

The appearance of the figure is highly sensitive to the ratio a/b. For a ratio of 1, the figure is an ellipse, with special cases including circles (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (a/b = 2, δ = π/2). Other ratios produce more complicated curves, which are closed only if a/b is rational. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures.

Lissajous figures where a = 1, b = N (N is a natural number) and

are Chebyshev polynomials of the first kind of degree N.