geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, Cayley's sextic (sextic of Cayley, Cayley's sextet) is a

plane curve In mathematics, a plane curve is a curve in a plane that may be 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 c ...

, a member of the sinusoidal spiral family, first discussed by

Colin Maclaurin Colin Maclaurin (; ; 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. ...

in 1718.

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...

was the first to study the curve in detail and Raymond Clare Archibald named the curve after him. 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 A pedal (from the Latin ''wikt:pes#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 us ...

(or ''roulette'') of a cardioid with respect to its cusp.

Equations of the curve

The equation of the curve in

polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...

is :''r'' = ''4a'' cos³(''θ''/3) In

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

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, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...

, period ,

\mathbb \rarr \mathbb^2

) by the equations: * ''x'' = cos³''t'' cos 3''t'' * ''y'' = cos³''t'' sin 3''t'' The node is at ''t'' = ±/3.

References

{{Reflist

External links

Mathworld
Sextic curves