Deltoid

$\begin{figure}\BoxedEPSF{deltoid.epsf scaled 500}\end{figure}$

A 3-cusped Hypocycloid, also called a Tricuspoid, which has $n\equiv a/b=3$ or , where is the Radius of the large fixed Circle and is the Radius of the small rolling Circle. The deltoid was first considered by Euler in 1745 in connection with an optical problem. It was also investigated by Steiner in 1856 and is sometimes called Steiner's Hypocycloid (MacTutor Archive). The equation of the deltoid is obtained by setting in the equation of the Hypocycloid, yielding the parametric equations

$\displaystyle x$	$\textstyle =$	$\displaystyle [{\textstyle{2\over 3}}\cos\phi-{\textstyle{1\over 3}}\cos(2\phi)]a = 2b\cos\phi+b\cos(2\phi)$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle [{\textstyle{2\over 3}}\sin\phi+{\textstyle{1\over 3}}\sin(2\phi)]a = 2b\sin\phi-b\sin(2\phi).$	(2)

$\begin{figure}\begin{center}\BoxedEPSF{DeltoidInfo.epsf scaled 700}\end{center}\end{figure}$

The Arc Length, Curvature, and Tangential Angle are

$\displaystyle s(t)$	$\textstyle =$	$\displaystyle 4\int_0^t \vert\sin({\textstyle{3\over 2}}t')\vert\,dt'={\textstyle{16\over 3}}\sin^2({\textstyle{3\over 4}}t)$	(3)
$\displaystyle \kappa(t)$	$\textstyle =$	$\displaystyle -{\textstyle{1\over 8}}\csc({\textstyle{3\over 2}}t)$	(4)
$\displaystyle \phi(t)$	$\textstyle =$	$\displaystyle -{\textstyle{1\over 2}}t.$	(5)

As usual, care must be taken in the evaluation of

for $t>2\pi/3$ . Since the form given above comes from an integral involving the Absolute Value of a function, it must be monotonic increasing. Each branch can be treated correctly by defining

$\begin{displaymath} n=\left\lfloor{3t\over 2\pi}\right\rfloor +1, \end{displaymath}$

(6)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function, giving the formula

$\begin{displaymath} s(t)=(-1)^{1+[n{\rm\ (mod\ 2)}]} {\textstyle{16\over 3}}\sin... ...2\over 3}}\left\lfloor{{\textstyle{1\over 2}}n}\right\rfloor . \end{displaymath}$