Lissajous curve

File:Lissajous-Figur 1 zu 3 (Oszilloskop).jpg

Lissajous figure on an oscilloscope

File:3D Lissajous figure (9, 4, 1).jpg

Lissajous figure in three dimensions

In mathematics, a Lissajous curve (Lissajous figure or Bowditch curve) is the graph of the system of parametric equations

<math>x=A\sin(at+\delta),\quad y=B\sin(bt),</math>

which describes complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous 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 (natural number) and <math>\delta=\frac{N-1}{N}\frac{\pi}{2}</math> are Chebyshev polynomials of the first kind of degree N.

Lissajous figures are sometimes used in graphic design as logos. Examples include the logos of the Australian Broadcasting Corporation (a = 1, b = 3, δ = π/2) and the Lincoln Laboratory at MIT (a = 4, b = 3, δ = 0).

Prior to modern computer graphics, Lissajous curves were typically generated using an oscilloscope (as illustrated). Two phase-shifted sinusoid inputs are applied to the oscilloscope in X-Y mode and the phase relationship between the signals is presented as a Lissajous figure. Lissajous curves can also be traced mechanically by means of a harmonograph.

In oscilloscope we suppose x is CH1 and y is CH2, A is amplitude of CH1 and B is amplitude of CH2, a is frequency of CH1 and b is frequency of CH2, so <math>a/b</math> is a ratio of frequency of two channels, finally, <math>\delta</math> is the phase shift of CH1.

Below are some examples of Lissajous figures with δ = π/2, a odd, b even, |a − b| = 1.

• a = 1, b = 2 (1:2)

  a = 1, b = 2 (1:2)

• a = 3, b = 2 (3:2)

  a = 3, b = 2 (3:2)

• a = 3, b = 4 (3:4)

  a = 3, b = 4 (3:4)

• a = 5, b = 4 (5:4)

  a = 5, b = 4 (5:4)

• a = 5, b = 6 (5:6)

  a = 5, b = 6 (5:6)

• a = 9, b = 8 (9:8)

  a = 9, b = 8 (9:8)

Spirograph

Even though they look similar, Spirographs are different as they are generally enclosed by a circular boundary where a Lissajous curve is bounded by a rectangle (±A, ±B).

See also

• Rose curve

External links

• Interactive Java Tutorial: Lissajous Figures on Oscilloscope National High Magnetic Field Laboratory
• Lissajous Curve at Mathworld
• Animated Lissajous figures in Java
• About the Australian Broadcasting Corporation logo
• Free tool QLiss3D that displays Lissajous figures in three dimensions
• A free Javascript tool for generating Lissajous curves
• Lissajous Curves: an interactive applet showing how to trace Lissajous curves in 2D. Requires Java.
• A 3D Java applet showing how a Lissajous figure can be traced.
• Lissajous 3D: animated textured 3D Lissajous patterns, also Lissajous screen saver - for Windows