one signal against time) to plot the output of an LTI system against the input to the LTI system produces an ellipse that is a Lissajous figure for the special case of . The aspect ratio of the resulting ellipse is a function of the phase shift between the input and output, with an aspect ratio of 1 (perfect circle) corresponding to a phase shift of ±90° and an aspect ratio of ∞ (a line) corresponding to a phase shift of 0° or 180°. The figure below summarizes how the Lissajous figure changes over different phase shifts. The phase shifts are all negative so that delay semantics can be used with a causal LTI system (note that −270° is equivalent to +90°). The arrows show the direction of rotation of the Lissajous figure. In engineering A Lissajous curve is used in experimental tests to determine if a device may be properly categorized as a memristor. It is also used to compare two different electrical signals: a known reference signal and a signal to be tested. In popular culture In motion pictures Lissajous figures were sometimes displayed on oscilloscopes meant to simulate high-tech equipment in science-fiction TV shows and movies in the 1960s and 1970s. The title sequence by John Whitney for Alfred Hitchcock's 1958 feature film Vertigo is based on Lissajous figures. Company logos Lissajous figures are sometimes used in graphic design as logos. Examples include: The Australian Broadcasting Corporation (, , ) The Lincoln Laboratory at MIT (, , ) The University of Electro-Communications, Japan (, , ). Disney's Movies Anywhere streaming video application uses a stylized version of the curve Facebook's rebrand into Meta Platforms is also a Lissajous Curve, echoing the shape of a capital letter M (, , ). In modern art The Dadaist artist Max Ernst painted Lissajous figures directly by swinging a punctured bucket of paint over a canvas. See also Lissajous orbit Blackburn pendulum Lemniscate of Gerono Spirograph Notes External links Lissajous Curve at Mathworld Interactive demos 3D Java applets depicting the construction of Lissajous curves in an oscilloscope: Tutorial from the NHMFL Physics applet by Chiu-king Ng Detailed Lissajous figures simulation Drawing Lissajous figures with interactive sliders in Javascript Lissajous Curves: Interactive simulation of graphical representations of musical intervals and vibrating strings Interactive Lissajous curve generator – Javascript applet using JSXGraph Animated Lissajous figures Wolfram Mathematica - Lissajous figures with interactive sliders