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\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 $3/2$, where $a$ is the Radius of the large fixed Circle and $b$ 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 $n=3$ 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 $s(t)$ 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
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
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} (7)

The total Arc Length is computed from the general Hypocycloid equation

s_n={8a(n-1)\over n}.
\end{displaymath} (8)

With $n=3$, this gives
s_3={\textstyle{16\over 3}} a.
\end{displaymath} (9)

The Area is given by
A_n = {(n-1)(n-2)\over n^2} \pi a^2
\end{displaymath} (10)

with $n=3,$
A_3 = {\textstyle{2\over 9}} \pi a^2.
\end{displaymath} (11)

The length of the tangent to the tricuspoid, measured between the two points $P$, $Q$ in which it cuts the curve again, is constant and equal to $4a$. If you draw Tangents at $P$ and $Q$, they are at Right Angles.


Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 53, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 131-135, 1972.

Lee, X. ``Deltoid.''

Lockwood, E. H. ``The Deltoid.'' Ch. 8 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 72-79, 1967.

Macbeth, A. M. ``The Deltoid, I.'' Eureka 10, 20-23, 1948.

Macbeth, A. M. ``The Deltoid, II.'' Eureka 11, 26-29, 1949.

Macbeth, A. M. ``The Deltoid, III.'' Eureka 12, 5-6, 1950.

MacTutor History of Mathematics Archive. ``Tricuspoid.''

Yates, R. C. ``Deltoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 71-74, 1952.

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© 1996-9 Eric W. Weisstein