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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 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 ±3⁄8√3a

The curve is the pedal curve (or roulette) of a cardoid with respect to its cusp.[1]

Equations of the curve

The equation of the curve in polar coordinates is[1][2]

*r*= 4*a*cos^{3}(*θ*/3)

One form of the Cartesian equation is[1][3]

4(*x*^{2} + *y*^{2} − *ax*)^{3} = 27*a*^{2}(*x*^{2} + *y*^{2})^{2} .

Cayley's sextic may be parametrised (as a periodic function, period π ℝ→ℝ2) by the equations Mathworld

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