any non-zero produces a figure that appears to be rotated, either as a left–right or an up–down rotation (depending on the ratio ). Lissajous figures where , ( is a natural number) and are Chebyshev polynomials of the first kind of degree . This property is exploited to produce a set of points, called Padua points, at which a function may be sampled in order to compute either a bivariate interpolation or quadrature of the function over the domain . The relation of some Lissajous curves to Chebyshev polynomials is clearer to understand if the Lissajous curve which generates each of them is expressed using cosine functions rather than sine functions. Examples The animation shows the curve adaptation with continuously increasing fraction from 0 to 1 in steps of 0.01 (). Below are examples of Lissajous figures with an odd natural number , an even natural number , and . Generation Prior to modern electronic equipment, Lissajous curves could be generated mechanically by means of a harmonograph. Practical application Lissajous curves can also be generated using an oscilloscope (as illustrated). An octopus circuit can be used to demonstrate the waveform images on an oscilloscope. Two phase-shifted sinusoid inputs are applied to the oscilloscope in X-Y mode and the phase relationship between the signals is presented as a Lissajous figure. In the professional audio world, this method is used for realtime analysis of the phase relationship between the left and right channels of a stereo audio signal. On larger, more sophisticated audio mixing consoles an oscilloscope may be built-in for this purpose. On an oscilloscope, we suppose is CH1 and is CH2, is the amplitude of CH1 and is the amplitude of CH2, is the frequency of CH1 and is the frequency of CH2, so is the ratio of frequencies of the two channels, and is the phase shift of CH1. A purely mechanical application of a Lissajous curve with , is in the driving mechanism of the Mars Light type of oscillating beam lamps popular with railroads in the mid-1900s. The beam in some versions traces out a lopsided figure-8 pattern on its side. Application for the case of When the input to an LTI system is sinusoidal, the output is sinusoidal with the same frequency, but it may have a different amplitude and some phase shift. Using an oscilloscope that can plot one signal against another (as opposed to