Lissajous figure on an oscilloscope- the shape of the ABC logo

Lissajous figure in three dimensions

In mathematics, a Lissajous curve (Lissajous figure or Bowditch curve) is the graph of the system of parametric equations ImageMetadata File history File links Lissajous-Figur_1_zu_3_(Oszilloskop). ... ImageMetadata File history File links Lissajous-Figur_1_zu_3_(Oszilloskop). ... A Tektronix model 475A portable analogue oscilloscope, a very typical instrument of the late 1970s. ... The Australian Broadcasting Corporation or ABC (formerly the Australian Broadcasting Commission) is Australias national non-profit public broadcaster. ... Image File history File links 3D_Lissajous_figure_(9,_4,_1). ... Image File history File links 3D_Lissajous_figure_(9,_4,_1). ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

which describes complex harmonic motion. This family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail by Jules Antoine Lissajous in 1857. Complex harmonic motion is the superposition — linear combination — of several simultaneous simple harmonic motions. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Memorial statue for Nathaniel Bowditch, in Mount Auburn Cemetery, Cambridge, Massachusetts. ... The Battle of New Orleans 1815 was a common year starting on Sunday (see link for calendar). ... Jules Antoine Lissajous (March 4, 1822 - June 24, 1880) was a French mathematician, after whom Lissajous figures are named. ...

The appearance of the figure is highly sensitive to the ratio a/b. For a ratio of 1, the figure is an ellipse, with special cases including circles (A = B, δ = π/2 radians) and lines (δ = 0). Another simple Lissajous figure is the parabola (a/b = 2, δ = π/2). Other ratios produce more complicated curves, which are closed only if a/b is rational. The visual form of these curves is often suggestive of a three-dimensional knot, and indeed the many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures. The ellipse and some of its mathematical properties. ... For information on the geometric shape, see circle and list of circle topics. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... The radian is a unit of plane angle. ... Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... It has been suggested that Analyzing the parabola be merged into this article or section. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Categories: Math stubs | Knot theory ...

Lissajous figures are sometimes used in graphic design as logos. Examples include the logos of the Australian Broadcasting Corporation (a = 1, b = 3, δ = π/2) and the Lincoln Laboratory at MIT (a = 8, b = 6, δ = 0). Saul Basss poster for the film The Man with the Golden Arm - a highly regarded work of graphic design. ... This article or section should include material from logo design, discuss it at Talk:Logo design A logotype, commonly known as a logo, is the graphic element of a trademark or brand, which is set in a special typeface/font, or arranged in a particular, but legible, way. ... The Australian Broadcasting Corporation or ABC (formerly the Australian Broadcasting Commission) is Australias national non-profit public broadcaster. ... MIT Lincoln Laboratory, also known as Lincoln Lab, is a federally funded research and development center managed by the Massachusetts Institute of Technology and funded by the United States Department of Defense. ... Mapúa Institute of Technology (MIT, MapúaTech or simply Mapúa) is a private, non-sectarian, Filipino tertiary institute located in Intramuros, Manila. ...

Prior to modern computer graphics, Lissajous curves were typically generated using an oscilloscope (as illustrated). 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. Lissajous curves can also be traced mechanically by means of a harmonograph. A Tektronix model 475A portable analogue oscilloscope, a very typical instrument of the late 1970s. ... A harmonograph is a mechanical apparatus that employs pendulums to create a geometric image. ...

In oscilloscope we suppose x is CH1 and y is CH2, A is amplitude of CH1 and B is amplitude of CH2, a is frequency of CH1 and b is frequency of CH2, so a / b is a ratio of frequency of two channels, finally, δ is phase shift of sin curve of CH1.

Below are some examples of Lissajous figures with δ = π/2, a odd, b even, |a − b| = 1.

a = 1, b = 2 (1:2) Image File history File links Lissajous_curve_1by2. ...	a = 3, b = 2 (3:2) Image File history File links Lissajous_curve_3by2. ...	a = 3, b = 4 (3:4) Image File history File links Lissajous_curve_3by4. ...	a = 5, b = 4 (5:4) Image File history File links Lissajous_curve_5by4. ...
a = 5, b = 6 (5:6) Image File history File links Lissajous_curve_5by6. ...	a = 9, b = 8 (9:8) Image File history File links Lissajous_curve_9by8. ...

Below figure shows the relationship of frequency ratio and phase shift

Image File history File links Lissajous. ...

Spirograph

Even though they look similar, Spirographs are different as they are generally enclosed by a circular boundary where a Lissajous curve is bounded by a square (±1, ±1). Several Spirograph designs drawn with a Spirograph set. ...

External links

• Lissajous Curve at Mathworld
• Animated Lissajous figures in Java
• About the Australian Broadcasting Corporation logo
• About the MIT Lincoln Laboratory logo
• Free tool QLiss3D that displays Lissajous figures in three dimensions
• Lissajous Curves - An interactive applet showing how to trace Lissajous curves in 2D. Requires Java.
• A 3D Java applet showing how a Lissajous figure be traced.