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Cayley's Sextic
In geometry, Cayley's sextic (sextic of Cayley, Cayley's sextet) is a plane curve, a member of the sinusoidal spiral family, first discussed by Colin Maclaurin in 1718. Arthur Cayley was the first to study the curve in detail and it was named after him in 1900 by Raymond Clare Archibald. The curve is symmetric about the ''x''-axis (''y'' = 0) and self-intersects at ''y'' = 0, ''x'' = −''a''/8. Other intercepts are at the origin, at (''a'', 0) and with the ''y''-axis at ± ''a'' The curve is the pedal curve (or ''roulette'') of a cardioid with respect to its cusp. Equations of the curve The equation of the curve in polar coordinates is :''r'' = ''4a'' cos3(''θ''/3) In Cartesian coordinates the equation is :4(''x''2 + ''y''2 − (''a''/4)''x'')3 = 27(''a''/4)2(''x''2 + ''y''2)2 . Cayley's sextic may be parametrised (as a periodic function A periodic function is a functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Sextic 001
Cayley may refer to: __NOTOC__ People * Cayley (surname) * Cayley Illingworth (1759–1823), Anglican Archdeacon of Stow * Cayley Mercer (born 1994), Canadian women's ice hockey player Places * Cayley, Alberta, Canada, a hamlet * Mount Cayley, a volcano in southwestern British Columbia, Canada * Cayley Glacier, Graham Land, Antarctica * Cayley (crater), a lunar crater Other uses * Cayley baronets, a title in the Baronetage of England * Cayley computer algebra system, designed to solve mathematical problems, particularly in group theory See also * W. Cayley Hamilton (died 1891), Canadian barrister and politician * Caylee (name), given name * Cèilidh, traditional Scottish or Irish social gathering * Kaylee, given name * Kaley (other) * Kayleigh (other) " Kayleigh" is a song by the British neo-progressive rock band Marillion. Kayleigh may also refer to: People *Kaylee, given name, including list of people named Kayleigh *Layla Kayleigh (born 1985), British-Ameri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. Symbolic representation A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form f(x,y)=0 for some specific function ''f''. If this equation can be solved explicitly for ''y'' or ''x'' – that is, rewritten as y=g(x) or x=h(y) for specific function ''g'' or ''h'' – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form (x,y)=(x(t), y(t)) for specific functions x(t) and y(t). Plane curves can sometimes also be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sinusoidal Spiral
In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates :r^n = a^n \cos(n \theta)\, where is a nonzero constant and is a rational number other than 0. With a rotation about the origin, this can also be written :r^n = a^n \sin(n \theta).\, The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including: * Rectangular hyperbola () * Line () * Parabola () * Tschirnhausen cubic () * Cayley's sextet () * Cardioid () * Circle () * Lemniscate of Bernoulli () The curves were first studied by Colin Maclaurin. Equations Differentiating :r^n = a^n \cos(n \theta)\, and eliminating ''a'' produces a differential equation for ''r'' and θ: :\frac\cos n\theta + r\sin n\theta =0. Then :\left(\frac,\ r\frac\right)\cos n\theta \frac = \left(-r\sin n\theta ,\ r \cos n\theta \right) = r\left(-\sin n\theta ,\ \cos n\theta \right) w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for being the youngest professor. The Maclaurin series, a special case of the Taylor series, is named after him. Owing to changes in orthography since that time (his name was originally rendered as M'Laurine), his surname is alternatively written MacLaurin. Early life Maclaurin was born in Kilmodan, Argyll. His father, John Maclaurin, minister of Glendaruel, died when Maclaurin was in infancy, and his mother died before he reached nine years of age. He was then educated under the care of his uncle, Daniel Maclaurin, minister of Kilfinan. A child prodigy, he entered university at age 11. Academic career At eleven, Maclaurin, a child prodigy at the time, entered the University of Glasgow. He graduated Master of Arts three years later by defending ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek language, Greek, French language, French, German language, German, and Italian language, Italian, as well as mathematics. He worked as a lawyer for 14 years. He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of a group (mathematics), group in the modern way—as a set with a Binary function, binary operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant permutation groups. Cayley tables and Cayley graphs as well as Cayle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raymond Clare Archibald
Raymond Clare Archibald (7 October 1875 – 26 July 1955) was a prominent Canadian-American mathematician. He is known for his work as a historian of mathematics, his editorships of mathematical journals and his contributions to the teaching of mathematics. Biography Raymond Clare Archibald was born in South Branch, Stewiacke, Nova Scotia on 7 October 1875. He was the son of Abram Newcomb Archibald (1849–1883) and Mary Mellish Archibald (1849–1901). He was the fourth cousin twice removed of the famous Canadian-American astronomer and mathematician Simon Newcomb. Archibald graduated in 1894 from Mount Allison College with B.A. degree in mathematics and teacher's certificate in violin. After teaching mathematics and violin for a year at the Mount Allison Ladies' College he went to Harvard where he received a B.A. 1896 and a M.A. in 1897. He then traveled to Europe where he attended the Humboldt University of Berlin during 1898 and received a Ph.D. cum laude from the Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Y-intercept
In analytic geometry, using the common convention that the horizontal axis represents a variable ''x'' and the vertical axis represents a variable ''y'', a ''y''-intercept or vertical intercept is a point where the graph of a function or relation intersects the ''y''-axis of the coordinate system. As such, these points satisfy ''x'' = 0. Using equations If the curve in question is given as y= f(x), the ''y''-coordinate of the ''y''-intercept is found by calculating f(0). Functions which are undefined at ''x'' = 0 have no ''y''-intercept. If the function is linear and is expressed in slope-intercept form as f(x)=a+bx, the constant term a is the ''y''-coordinate of the ''y''-intercept. Multiple y-intercepts Some 2-dimensional mathematical relationships such as circles, ellipses, and hyperbolas can have more than one ''y''-intercept. Because functions associate ''x'' values to no more than one ''y'' value as part of their definition, they can have at most one ''y''-intercept. x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pedal Curve
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is used to control playback of voice dictations Geometry * Pedal curve, a curve derived by construction from a given curve * Pedal triangle, a triangle obtained by projecting a point onto the sides of a triangle Music Albums * ''Pedals'' (Rival Schools album) * ''Pedals'' (Speak album) Other music * Bass drum pedal, a pedal used to play a bass drum while leaving the drummer's hands free to play other drums with drum sticks, hands, etc. * Effects pedal, a pedal used commonly for electric guitars * Pedal keyboard, a musical keyboard operated by the player's feet * Pedal harp, a modern orchestral harp with pedals used to change the tuning of its strings * Pedal point, a type of nonchord tone, usually in the bass * Pedal tone, a fundamental tone played ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardioid
In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle. The name was coined by de Castillon in 1741 but the cardioid had been the subject of study decades beforehand.Yates Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid (any 2d plane containing the 3d straight line of the microphone body). In three dimensions, the cardioid is shaped like an apple centred around the microphone which is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the ''pole'', and the ray from the pole in the reference direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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