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Homesemicubical parabola

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# semicubical parabola

The curve $y=x^{3}$ is sometimes called a cubic(al) parabola, as $y=x^{2}$ a parabola. Thus there is reason to call the curve

$y\;=\;x^{{\frac{3}{2}}}\;=\;x\sqrt{x}$ |

a semicubical parabola. Since this equation implies that

$\displaystyle y^{2}\;=\;x^{3},$ | (1) |

which is satisfied as well by negative real numbers $y$, it is natural to join to the semicubical parabola also points below the $x$-axis. Therefore the equation (1) can be equivalently written as

$\displaystyle y\;=\;\pm{x}^{{\frac{3}{2}}}\;=\;\pm{x}\sqrt{x}.$ | (2) |

According to (1) or (2), the semicubical parabola has symmetry about the $x$-axis. Because the positive branch of (2) forms a strictly increasing and the negative branch a strictly decreasing power function for $x\leqq 0$, the graph of (2) has an ordinary cusp in the origin.

The arc length of the semicubical parabola from the origin to the abscissa $x$ is

$s(x)\;=\;\int_{0}^{x}\sqrt{1+\!\left(\frac{d}{dt}t^{\frac{3}{2}}\right)^{2}}\,% dt\;=\;\frac{(4\!+\!9x)\sqrt{4\!+\!9x}-8}{27},$ |

i.e. an algebraic expression in $x$. Actually, the semicubical parabola is historically the first algebraic curve (after the straight line) having an algebraic arc length.

Making a squeezing of the plane, one can write a more general equation

$\displaystyle ay^{2}\;=\;x^{3}$ | (3) |

of the semicubical parabola; here $a$ is a positive constant.

The semicubic parabola is also the evolute of a parabola; e.g. the equation

$3y^{2}\;=\;(x-\frac{1}{2})^{3}$ |

represents the evolute of the parabola $y^{2}=x$ (cf. determining envelope).

## Mathematics Subject Classification

53A04*no label found*

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