{{Short description|Plane curve}} thumb|300x300px 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 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 ±{{frac|3|8}}{{sqrt|3}}''a''
The curve is the pedal curve (or ''roulette'') of a cardioid with respect to its cusp.<ref name=Law178>{{cite book | first=J. Dennis | last=Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | page=[https://archive.org/details/catalogofspecial00lawr/page/178 178] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/178 }} </ref>
==Equations of the curve== The equation of the curve in polar coordinates is<ref name=Law178/><ref>{{Cite book|title=Academic Press Dictionary of Science and Technology|author=Christopher G. Morris|page=381}}</ref>
:''r'' = ''4a'' cos<sup>3</sup>(''θ''/3)
In Cartesian coordinates the equation is<ref name=Law178/><ref>{{Cite book|page=62|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|author=David Darling|publisher=John Wiley and Sons|date=28 October 2004|isbn= 9780471667001}}</ref>
:4(''x''<sup>2</sup> + ''y''<sup>2</sup> − (''a''/4)''x'')<sup>3</sup> = 27(''a''/4)<sup>2</sup>(''x''<sup>2</sup> + ''y''<sup>2</sup>)<sup>2</sup> .
Cayley's sextic may be parametrised (as a periodic function, period {{pi}}, <math>\mathbb{R} \rarr \mathbb{R}^2</math>) by the equations:
* ''x'' = cos<sup>3</sup>''t'' cos 3''t'' * ''y'' = cos<sup>3</sup>''t'' sin 3''t''
The node is at ''t'' = ±{{pi}}/3.<ref>{{Cite book|title=Elementary Geometry of Differentiable Curves: An Undergraduate Introduction|author=C. G. Gibson|publisher=Cambridge University Press|year=2001|isbn= 9780521011075}}</ref>
==References== {{Reflist}}
==External links== * [http://mathworld.wolfram.com/CayleysSextic.html Mathworld]
Category:Sextic curves