Lissajous Knot With Parameters (4, 5, 41, 0
Lissajous may refer to * Jules Antoine Lissajous (1822–1880), French mathematician ** Lissajous curve (or figure, or spiral), a mathematical figure showing a type of harmonic motion ** Lissajous knot, in knot theory ** Lissajous orbit In orbital mechanics, a Lissajous orbit (), named after Jules Antoine Lissajous, is a quasi-periodic orbital trajectory that an object can follow around a Lagrangian point of a three-body system without requiring any propulsion. Lyapunov orbits ...

Jules Antoine Lissajous
Jules Antoine Lissajous (; 4 March 1822 in Versailles – 24 June 1880 in Plombières-les-Dijon) was a French physicist, after whom Lissajous figures are named. Among other innovations, Lissajous invented the Lissajous apparatus, a device that creates the figures that bear his name. In it, a beam of light is bounced off a mirror attached to a vibrating tuning fork, and then reflected off a second mirror attached to a perpendicularly oriented vibrating tuning fork (usually of a different pitch, creating a specific harmonic interval), onto a wall, resulting in a Lissajous figure. This led to the invention of other apparatus such as the harmonograph. See also *Lissajous curve *Lissajous orbit In orbital mechanics, a Lissajous orbit (), named after Jules Antoine Lissajous, is a quasi-periodic orbital trajectory that an object can follow around a Lagrangian point of a three-body system without requiring any propulsion. Lyapunov orbits ... References External links * 1822 ...
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Lissajous Curve
A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y directions of different angular frequency (''a'' and ''b).'' The resulting family of curves was investigated by Nathaniel Bowditch in 1815, and later in more detail in 1857 by Jules Antoine Lissajous (for whom it has been named). Such motions may be considered as a particular of kind of complex harmonic motion. The appearance of the figure is sensitive to the ratio . For a ratio of 1, when the frequencies match a=b, the figure is an ellipse, with special cases including circles (, radians) and lines (). A small change to one of the frequencies will mean the x oscillation after one cycle will be slightly out of synchronization with the y motion and so the ellipse will fail to close and trace a curve slightly adjacent during the next orbit sho ...
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Lissajous Knot
In knot theory, a Lissajous knot is a knot defined by parametric equations of the form :x = \cos(n_x t + \phi_x),\qquad y = \cos(n_y t + \phi_y), \qquad z = \cos(n_z t + \phi_z), where n_x, n_y, and n_z are integers and the phase shifts \phi_x, \phi_y, and \phi_z may be any real numbers. The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves. Replacing the cosine function in the parametrization by a triangle wave transforms every Lissajous knot isotopically into a billiard curve inside a cube, the simplest case of so-called ''billiard knots''. Billiard knots can also be studied in other domains, for instance in a cylinder or in a (flat) solid torus ( Lissajous-toric knot). Form Because a knot cannot be self-intersecting, the three integers n_x, n_y, n_z must be pairwise relatively prime, and none of the quantities :n_x \phi_y - n_y \phi ...
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