The Beauty of Math in Science - Lissajous Curve
Lissajous curve, also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equations: x = A.sin(a.t + δ) and y = B.cos(bt)
The appearance of the figure is highly sensitive to the ratio a/b - Image 3 (3/2, 3/4 and 5/4). 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/4). 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.
Visually, the ratio a/b determines the number of “lobes” of the figure. For example, a ratio of 3/1 or 1/3 produces a figure with three major lobes (see image). The ratio A/B determines the relative width-to-height ratio of the curve. For example, a ratio of 2/1 produces a figure that is twice as wide as it is high. Finally, the value of δ determines the apparent “rotation” angle of the figure, viewed as if it were actually a three-dimensional curve. For example, δ=0 produces x and y components that are exactly in phase, so the resulting figure appears as an apparent three-dimensional figure viewed from straight on (0°). In contrast, any non-zero δ produces a figure that appears to be rotated, either as a left/right or an up/down rotation (depending on the ratio a/b).
See more at source: Lissajous curve.
Images: 3D Lissajous curve - Lissajous curve - How to Make a Three-Pendulum Rotary Harmonograph by Karl Sims.