Cayley's sextic explained

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.^[1]

Equations of the curve

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

r = 4a cos³(θ/3)

In Cartesian coordinates the equation is^{[1] [3]}

4(x² + y² − (a/4)x)³ = 27(a/4)²(x² + y²)² .

Cayley's sextic may be parametrised (as a periodic function, period,

R \rarr R ^{2} ) by the equations:

• x = cos³t cos 3t
• y = cos³t sin 3t

The node is at t = ±/3.^[4]

External links

• Mathworld

Notes and References

1. Book: Lawrence, J. Dennis . A catalog of special plane curves . Dover Publications . 1972 . 0-486-60288-5 . 178 . registration .
2. Book: Academic Press Dictionary of Science and Technology. Christopher G. Morris. 381.
3. Book: 62. The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. David Darling. John Wiley and Sons. 28 October 2004. 9780471667001.
4. Book: Elementary Geometry of Differentiable Curves: An Undergraduate Introduction. C. G. Gibson. Cambridge University Press. 2001. 9780521011075.