Cayley´s sextic has been studied for the first time by MacLaurin (1718). Later Arthur (1821-1895) took a closer look to the curve. Archibald named the curve after Cayley in a paper classifying some curves, a paper that was published in 1900 in Strasbourg.We call the origin in the curve's polar equation the focus of the curve. Some relationships with other curves: - it is a roulette, formed by a cardioid rolling over another cardioid with the same size
- the curve is the pedal of the cardioid
- it is the involute of the nephroid
- it is a polar inverse of the Tschirnhausen's cubic
The area of the curve with respect to $a$ is broken down into two parts, the inner loop and the outer loop. The area of the inner loop could be found by $\displaystyle A_{inner}=\frac{1}{2}\left ( 5 \pi -9\sqrt{3} \right )a^2\approx \left(0.05975299... \right)a^2$. The area of the outer loop could be found by $\displaystyle A_{outer}=\left ( 5 \pi -\frac{9}{2}\sqrt{3} \right )a\approx \left (23.50219... \right )a$. The arc length of the whole curve with respect to $a$ can be found by $\displaystyle s=6\pi a$. **Click Read More to see the interactive graphic** | Equations: Cartesian Equation: $\displaystyle 4 \left ( x^2+y^2-ax \right )^3=27a^2\left ( x^2 +y^2\right )^2$ Polar Equation: $\displaystyle r=4a \cos^3\left ( \frac{\theta}{3} \right )$ Parametric Equations: $\displaystyle \begin{pmatrix} x\left ( t \right )\\ y\left ( t \right ) \end{pmatrix} = \begin{pmatrix} 4a\cos^3 \left ( \frac{t}{2} \right ) \cos\left ( \frac{3t}{2} \right )\\ 4a\cos^3 \left ( \frac{t}{2} \right ) \sin\left ( \frac{3t}{2} \right ) \end{pmatrix}$ Graph: |

**Interactive Graphic:**

**References:**

- "Cayley's Sextic."
*Cayleys*. N.p., Jan. 1997. Web. 21 Dec. 2013. - Gu, Weiqing. "Cayley's Sextic."
*Cayley's Sextic*. Harvey Mudd College, n.d. Web. 21 Dec. 2013. - Wassenaar, Jan. "Cayley's Sextic."
*Cayley's Sextic*. N.p., n.d. Web. 21 Dec. 2013. - Weisstein, Eric W. "Cayley's Sextic."
*Wolfram MathWorld*. Wolfram Research, Inc., 16 Mar. 2008. Web. 21 Dec. 2013. - Weisstein, Eric W. "Roulette."
*Wolfram MathWorld*. Wolfram Research, Inc., 8 Oct. 2003. Web. 21 Dec. 2013. - Wolfram|Alpha. "Cayley's Sextic."
*Wolfram|Alpha*. Wolfram Research, Inc, 21 Dec. 2013. Web. 21 Dec. 2013.