# 7. Lissajous Figures

by M. Bourne

Lissajous figure

*x* = sin(*t*/18),

*y* = cos(2*t*/20)

We can obtain very interesting graphs when each of the
*x-* and *y-* coordinates are given as functions of
*t*. In this case, we have **parametric equations.** (We see another example of parametric equations later in the applications of differentiation section.)

**Lissajous Figures** are a special case of parametric equations, where *x* and *y* are in the following form:

`x = A\ sin(at + δ)`

`y = B\ sin(bt + γ)`

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[These can also be written in terms of cosine expressions, or a combination of sin and cos, since we can shift sine onto cosine easily. See Graphs of y = a sin(bx + c).]

Lissajous curves can be seen on **oscilloscopes** and are the
result of combining 2 trigonometric curves at right angles.

### Example 1

### Need Graph Paper?

Sketch the graph of the parametric equation where:

x= 2 cost

y= cos(t+ 4)

Answer

We need to set up a table of values to see what is happening. We give each point a "point number" so it is easier to understand when we graph the curve.

`t` | `0` | `π/4` | `π/2` | `(3π)/4` | `π` | `(5π)/4` | `(3π)/2` | `(7π)/4` | `2π` |

`x` | `2` | `1.4` | `0` | `-1.4` | `-2` | `-1.4` | `0` | `1.4` | `2` |

`y` | `-0.6` | `0.1` | `0.7` | `1.0` | `0.7` | `-0.1` | `-0.8` | `-1.0` | `-0.7` |

Pt no. |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Here is the resulting curve, with the numbered points included. Point 1 is actually equivalent to Point 9, since we have done one complete cycle of the ellipse as *t* goes from 0 to 2π.

Lissajous figure: Ellipse

*x* = 2 cos(*t*),

*y* = cos(*t* + 4)

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## Common Lissajous Curves

Lissajous curves take certain common shapes depending on the values of the variables in the expressions

x=Asin(at+δ) and

y=Bsin(bt+γ)

In Example 1, we saw that the curve was an ellipse. If *A* ≠ *B* and *a *=* b*, we obtain an ellipse. (See more on the Ellipse.)

In the example in Curvilinear Motion, the Lissajous figure is a circle. If *A *=* B* and *a *=* b *= 1, we will get a circle.

Example 3 on this page is part of a parabola. We can also obtain a straight line as well.

### Example 2 - ABC Logo

The Australian Broadcasting Corporation is a (mostly!) high quality public television and radio network. The ABC logo is a Lissajous figure. The parametric equations that describe the logo are:

x= sint

y= cos 3t

The graph is as follows:

ABC logo

*x* = sin(*t*),

*y* = cos(3*t*)

For more information by the ABC, see The ABC's of Lissajous Figures.

### Example 3

Sketch the graph of the parametric function:

`x = cos(t + π/4)`

`y = sin\ 2t`

Answer

We draw a table of values:

`t` | `0` | `π/4` | `π/2` | `(3π)/4` | `π` | `(5π)/4` | `(3π)/2` | `(7π)/4` | `2π` |

`x` | `0.7` | `0` | `-0.7` | `-1` | `-0.7` | `0` | `0.7` | `1` | `0.7` |

`y` | `0` | `1` | `0` | `-1` | `0` | `1` | `0` | `-1` | `0` |

Pt no . |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

The sketch is:

Lissajous figure: Parabola (part)

*x* = cos(*t* + π/4),

*y* = sin(2*t*)

This curve is part of a parabola.

Of course, it would have been better to start our values from `t = (3π)/4` (Point 4) and only use Points 4, 5, 6, 7, and 8, but often you don't know that until you have drawn it.

### Example 4 - Mathematical heart

This last one's not actually a Lissajous figure, but like such curves, it's made of parametric equations involving trigonometric functions.

The parametric equations are:

*x* = 5 sin^{3}*t*,

*y* = 4 cos(*t*) − 1.3 cos(2*t*) − 0.6 cos(3*t*) − 0.2 cos(4*t*)

Mathematical heart

*x* = 5 sin^{3}*t*,

*y* = 4 cos(*t*) − 1.3 cos(2*t*) − 0.6 cos(3*t*) − 0.2 cos(4*t*)

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