Semicubical parabola

• Any semicubical parabola ${\displaystyle (t^{2},at^{3})}$ is similar to the semicubical unit parabola ${\displaystyle (u^{2},u^{3})}$ .

Proof: The similarity ${\displaystyle (x,y)\rightarrow (a^{2}x,a^{2}y)}$ (uniform scaling) maps the semicubical parabola ${\displaystyle (t^{2},at^{3})}$ onto the curve ${\displaystyle ((at)^{2},(at)^{3})=(u^{2},u^{3})}$ with ${\displaystyle u=at}$.

• The parametric representation ${\displaystyle (t^{2},at^{3})}$ is regular except at point ${\displaystyle (0,0)}$. At point ${\displaystyle (0,0)}$ the curve has a singularity (cusp).

The proof follows from the tangent vector ${\displaystyle (2t,3t^{2})^{T}}$. Only for ${\displaystyle t=0}$ this vector has zero length.

Tangent at a semicubical parabola

Differentiating the semicubical unit parabola ${\displaystyle \;y=\pm x^{\frac {3}{2}}\;}$ one gets at point ${\displaystyle (x_{0},y_{0})}$ of the upper branch the equation of the tangent:

• ${\displaystyle y={\frac {\sqrt {x_{0}}}{2}}{\Big (}3x-x_{0}{\Big )}\;.}$

This tangent intersects the lower branch at exactly one further point with coordinates [3]

• ${\displaystyle {\Big (}{\frac {x_{0}}{4}},-{\frac {y_{0}}{8}}{\Big )}\;.}$

(Proving this statement one should use the fact, that the tangent meets the curve at ${\displaystyle (x_{0},y_{0})}$ twice.)

Determining the arclength of a curve ${\displaystyle (x(t),y(t))}$ one has to solve the integral ${\displaystyle \int {\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt}$. For the semicubical parabola ${\displaystyle (t^{2},at^{3}),\;0\leq t\leq b\;,}$ one gets

${\displaystyle \int _{0}^{b}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt=\int _{0}^{b}t{\sqrt {4+9a^{2}t^{2}}}\;dt=\cdots ={\Big [}{\frac {1}{27a^{2}}}(4+9a^{2}t^{2})^{\frac {3}{2}}{\Big ]}_{0}^{b}\;.}$

(The integral can be solved by the substitution ${\displaystyle u=4+9a^{2}t^{2}}$.)

Example: For ${\displaystyle a=1}$ (semicubical unit parabola) and ${\displaystyle b=2}$, which means the length of the arc between the origin and point ${\displaystyle (4,8)}$, one gets the arc length ${\displaystyle 9.073\;.}$

• The evolute of the parabola ${\displaystyle (t^{2},t)}$ is a semicubical parabola shifted by 1/2 along the x-axis: ${\displaystyle ({\frac {1}{2}}+t^{2},{\frac {4}{{\sqrt {3}}^{3}}}t^{3})\;.}$

In order to get the representation of the semicubical parabola ${\displaystyle (t^{2},at^{3})}$ in polar coordinates, one determines the intersection point of the line ${\displaystyle y=mx}$ with the curve. For ${\displaystyle m\neq 0}$ there is one point different from the origin: ${\displaystyle ({\frac {m^{2}}{a^{2}}},{\frac {m^{3}}{a^{2}}})}$. This point has distance ${\displaystyle {\frac {m^{2}}{a^{2}}}{\sqrt {1+m^{2}}}}$ from the origin. With ${\displaystyle m=\tan \varphi }$ and ${\displaystyle \sec ^{2}\varphi =1+\tan ^{2}\varphi }$ ( see List of identities) one gets [4]

• ${\displaystyle r={\Big (}{\frac {\tan \varphi }{a}}{\Big )}^{2}\cdot \sec \varphi \;,\quad -\pi /2<\varphi <\pi /2\ .}$

Relation between a semicubical parabola and a cubic function (green)

Mapping the semicubical parabola ${\displaystyle (t^{2},t^{3})}$ by the projective map ${\displaystyle (x,y)\rightarrow ({\tfrac {x}{y}},{\tfrac {1}{y}})}$ (involutoric perspectivity with axis ${\displaystyle y=1}$ and center ${\displaystyle (0,-1)}$ ) yields ${\displaystyle ({\frac {1}{t}},{\frac {1}{t^{3}}})}$, hence the cubic function ${\displaystyle y=x^{3}}$. The cusp (origin) of the semicubical parabola is exchanged with the point at infinity of the y-axis.

This property can be derived, too, if one represents the semicubical parabola by homogeneous coordinates: In equation (A) the replacement ${\displaystyle x={\tfrac {x_{1}}{x_{3}}},\;y={\tfrac {x_{2}}{x_{3}}}}$ (the line at infinity has equation ${\displaystyle x_{3}=0}$.) and the multiplication by ${\displaystyle x_{3}^{3}}$ is performed. One gets the equation of the curve

• in homogeneous coordinates: ${\displaystyle \quad x_{3}x_{2}^{2}-x_{1}^{3}=0\;.}$

Choosing line ${\displaystyle x_{\color {red}2}=0}$ as line at infinity and introducing ${\displaystyle x={\tfrac {x_{1}}{x_{2}}},\;y={\tfrac {x_{3}}{x_{2}}}}$ yields the (affine) curve ${\displaystyle y=x^{3}\;.}$

An additional defining property of the semicubical parabola is that it is an isochrone curve, meaning that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods. In this way it is related to the tautochrone curve, for which particles at different starting points always take equal time to reach the bottom, and the brachistochrone curve, the curve that minimizes the time it takes for a falling particle to travel from its start to its end.