Lissajous curves

Let's imagine a curve drawn on a plane, in particular it can be a straight line or a circle. If we place a Cartesian coordinate system on this plane, we can try to describe this curve using the y(x) function. An example of such a function may be the equation y = x (it is a linear function). In other words, having such a function, we can calculate the value depending on the value of x.


However, we very often encounter situations where parametric equations are used to describe curves in space. For example, in kinematics, motion on a plane can be described by two equations determining the coordinates of a moving point depending on time t. In other words, two dependencies are given: x(t) and y(t). Having such two parametric equations, we can determine the point at which a body moving in the plane is located for any moment of time t. For example: let x(t) = t and y(t) = 2•t. These equations show that the point moves in the Y direction twice as fast as in the direction indicated by the X axis. In this example, you can try simple transformations:

x = t     y = 2t     therefore: y = 2x

Let us now consider the motion of a point in a plane described by the following two parametric equations:

x(t) = x0 • sin( 2π•fx•t + Φx)     y(t) = y0 • sin( 2π•fy•t + Φy)

Along the X axis, the point oscillates with frequency fx and amplitude x0, and along the Y axis the point vibrates with frequency fy and amplitude y0. The quantities Φx and Φy are the so-called angular phase shifts.
Therefore, the above two parametric equations describe the motion of a point performing harmonic vibrations in two mutually perpendicular directions X,Y.
A Lissajous curve
is a curve showing the path of such a moving point in a plane.

Let us experimentally examine the path of a point whose motion is described by the parametric equations as above. Set the vibration parameters of the point along the X and Y axes. Then, on the graph below, the point will be in the initial position given by the equations:

x(t=0) = x0 • sin( Φx)     y(t=0) = y0 • sin( Φy)
Now you can initiate the point movement and track its path.

Amplitude x0 =
Freq. fx =   
Angular shift Φx = deg
Amplitude y0 =
Freq. fy =   
Angular shift Φy = deg


This is only a simulator, so do not expect to see a point move on the screen along the X and Y axes at the exact frequency you set. It is important that the curve along which the point moves is consistent with the set parameters.

In the formulas above, the arguments of the sine function, including the angular phase shifts Φx and Φy , must be expressed in radians (rad - arc measure of an angle). For the user's convenience, in the parameter setting section, the amounts of angular shifts are entered and expressed in degrees (deg - a degree measure of an angle).
Notice that:    360 deg = 2π rad.   The number π is Archimedes' constant. This constant has an approximate value π ≈ 3,14.
Below is a calculator for converting angle measures.

deg = 0.5π rad        π rad = 360 deg       

★ ★ ★

In an electronics laboratory, Lissajous curves can be generated on the screen of an oscilloscope operating in XY mode. In the past, this was often used to investigate phase or frequency differences between two electrical signals. One of the signals came from a laboratory generator and had well-defined parameters, in particular amplitude and frequency. The parameters of the second signal, which was the subject of the study, were determined from the shape of the obtained Lissajous curve.