# Lissajous Figure

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October 17, 2021 A lissajous figure is a curve formed by the trajectory of a point that simultaneously participates in two mutually perpendicular harmonics. The following expressions describe this figure: X a sin ⁡ ( a t + δ ) {\displaystyle xA\sin(at+\delta )} and y B sin ⁡ ( b t ) {\displaystyle \quad yB\sin(bt)} Lissajous figures are named after Jules Antoine Lissajous (1822-1880). He obtained the figures by successively reflecting light through two mirrors attached to two tuning forks at right angles to each other. Lissajous figures become visible on the screen of an oscilloscope, if one connects the inputs for the horizontal and vertical signals with sinusoidal voltages with frequencies in fixed ratios. With two mirror galvanometers they can be projected onto a plane; this principle is applied in laser shows. For frequency ratios that form simple fractions, easily recognizable lissajous figures arise, making it possible to interpret the relationship between the two frequencies. Because only still pictures appear on the screen if the frequency ratios are exactly integers, such as 2:1, 3:2, 4:3, two frequencies can be adjusted very accurately with respect to each other with the help of these figures. When the frequencies deviate slightly from this exact ratio, the difference frequency figure will change as if the relative phase of the two waves were changed. A simple lissajous figure is a circle. This occurs when both vibrations have the same frequency and amplitude, with a phase difference π/2. Unequal amplitudes yield an oval. A phase difference 0 produces a diagonal, straight line segment. If one changes the phase difference over time, the figure appears to revolve around its x-axis or y-axis. This creates an impression of a 3-dimensional figure.

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