Lissajous Figures

Definition

Lissajous figures are closed curves that are created by the parametric equations of two simple harmonic motions at different frequencies. These figures are named after the French physicist Jules Antoine Lissajous, who studied them extensively in the 19th century.

5 Must Know Facts For Your Next Test

1. Lissajous figures are created by the combination of two simple harmonic motions with different frequencies, amplitudes, and phase differences.
2. The shape of the Lissajous figure is determined by the ratio of the two frequencies and the phase difference between the two motions.
3. Lissajous figures can be used to visually represent the phase difference between two periodic signals, as well as to measure the frequency ratio of the two signals.
4. Lissajous figures have a wide range of applications, including in the field of electronics for the display of waveforms on oscilloscopes.
5. The parametric equations for a Lissajous figure are $x = A \cos(2\pi f_1 t + \phi_1)$ and $y = B \cos(2\pi f_2 t + \phi_2)$, where $A$ and $B$ are the amplitudes, $f_1$ and $f_2$ are the frequencies, and $\phi_1$ and $\phi_2$ are the phase differences of the two simple harmonic motions.

Review Questions

• Explain how the shape of a Lissajous figure is determined by the ratio of the two frequencies and the phase difference between the two motions.
  • The shape of a Lissajous figure is determined by the ratio of the frequencies of the two simple harmonic motions and the phase difference between them. If the frequency ratio is a rational number, the Lissajous figure will be a closed curve with a repeating pattern. The number of loops or lobes in the figure is determined by the numerator and denominator of the frequency ratio. The phase difference between the two motions affects the orientation and symmetry of the Lissajous figure, with different phase differences resulting in different shapes, such as ellipses, circles, and figure-eight patterns.
• Describe the parametric equations that define a Lissajous figure and explain the role of each parameter in the equation.
  • The parametric equations that define a Lissajous figure are $x = A \cos(2\pi f_1 t + \phi_1)$ and $y = B \cos(2\pi f_2 t + \phi_2)$, where $A$ and $B$ are the amplitudes of the two simple harmonic motions, $f_1$ and $f_2$ are their frequencies, and $\phi_1$ and $\phi_2$ are their phase differences. The frequency ratio $f_1/f_2$ determines the number of loops or lobes in the Lissajous figure, while the phase difference $\phi_1 - \phi_2$ affects the orientation and symmetry of the figure. The amplitudes $A$ and $B$ determine the size and proportions of the Lissajous figure.
• Analyze the practical applications of Lissajous figures, particularly in the field of electronics, and explain how they can be used to measure the frequency ratio and phase difference between two periodic signals.
  • Lissajous figures have a wide range of practical applications, particularly in the field of electronics. One of the most common uses is in oscilloscopes, where Lissajous figures are used to display the waveforms of two periodic signals and visually represent the phase difference between them. By adjusting the frequency and phase of the two signals, the shape of the Lissajous figure can be used to measure the frequency ratio and phase difference between the signals. This technique is useful for troubleshooting and analyzing the behavior of electronic circuits, as well as for measuring the characteristics of various periodic signals, such as those found in audio and communication systems. The versatility and intuitive nature of Lissajous figures make them a valuable tool in the field of electronics and beyond.

Related terms

Parametric Equations: Parametric equations are a way of representing a curve in the coordinate plane by using a parameter, typically denoted as 't', to define the x and y coordinates separately as functions of that parameter.

Simple Harmonic Motion: Simple harmonic motion is a type of periodic motion where the restoring force is proportional to the displacement from the equilibrium position, resulting in a sinusoidal oscillation.

Phase Difference: The phase difference is the difference in the phase angles of two or more periodic signals, which determines the shape of the Lissajous figure.