Neil's parable

Neil's parabolas for different values ​​of a .

The Neil parabola (named after the English mathematician William Neile ) or semicubic parabola is a special planar algebraic curve that is represented by an equation of form

• (A) ${\displaystyle \quad y^{2}-a^{2}x^{3}\;=\,0\;,\;a>0\;,}$

can be described. Solving for gives the explicit form ${\displaystyle y}$

• (E1) ${\displaystyle \quad y=\pm ax^{\frac {3}{2}}\;,\;x\geq 0\;,}$

which gives rise to the designation semicubic parabola .

(An ordinary parabola can be described by an equation .) ${\displaystyle y=ax^{2}}$

Solving (A) for gives the equation ${\displaystyle x}$

• (E2) ${\displaystyle \quad x={\Big (}{\frac {y}{a}}{\Big )}^{\color {red}{\frac {2}{3}}}\;,\quad y\in \mathbb {R} \;.}$

The first equation shows that

• (P) ${\displaystyle \quad x=t^{2}\;,\quad y=at^{3}\;,\quad t\in \mathbb {R} \;,}$

is a parametric representation of Neil's parabola.

William Neile first calculated the arc length of this curve , the so-called rectification , and made this known in 1657. Because of the problems with rectifying ellipses and parabolas, it was assumed at that time that the circle and the straight line were the only rectifiable algebraic curves.

Neil's parabola is rational , so there is a rational mapping with an inverse rational mapping that maps Neil's parabola to the projective straight line .

Properties of a Neil parabola

similarity

• Every Neil parabola is similar to Neil's unit parabola . ${\displaystyle (t^{2},at^{3})}$ ${\displaystyle (u^{2},u^{3})}$

Proof: The Homothecy (stretching at the origin) leads the semicubical parabola into the corner with about. ${\displaystyle (x,y)\rightarrow (a^{2}x,a^{2}y)}$ ${\displaystyle (t^{2},at^{3})}$ ${\displaystyle ((at)^{2},(at)^{3})=(u^{2},u^{3})}$ ${\displaystyle u=at}$

Singularity

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

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

Neil's parabola: tangent

Tangents

For Neil's unit parabola, differentiation results in the equation of the tangent at a point on the upper branch: ${\displaystyle \;y=\pm x^{\frac {3}{2}}\;}$ ${\displaystyle (x_{0},y_{0})}$

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

This tangent intersects the curve in exactly one further point of the lower branch with the coordinates

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

(When recalculating you should take into account that there is a double intersection of the tangent with the curve.) ${\displaystyle (x_{0},y_{0})}$

Arc length

In order to determine the arc length of a parameterized curve, one has to solve the indefinite integral . For Neil's parable is ${\displaystyle (x(t),y(t))}$ ${\displaystyle \int {\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt}$ ${\displaystyle (t^{2},at^{3}),\;0\leq t\leq b\;,}$

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

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

Example: For (Neil's unit parabola) and the upper limit , i.e. H. to the point is the length . ${\displaystyle a=1}$ ${\displaystyle b=2}$ ${\displaystyle (4,8)}$ ${\displaystyle 9{,}073}$

Evolute the unit parabola

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

Polar coordinates

To find the representation of Neil's parabola in polar coordinates, one intersects the straight line through the origin with the curve. For there is one from the zero point (tip) different point: . The distance from this point to the zero point is . With and results ${\displaystyle (t^{2},at^{3})}$ ${\displaystyle y=mx}$ ${\displaystyle m\neq 0}$ ${\displaystyle \left({\frac {m^{2}}{a^{2}}},{\frac {m^{3}}{a^{2}}}\right)}$ ${\displaystyle {\frac {m^{2}}{a^{2}}}{\sqrt {1+m^{2}}}}$ ${\displaystyle m=\tan \varphi }$ ${\displaystyle \sec ^{2}\varphi =1+\tan ^{2}\varphi }$

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

Neil's parabola and cubic parabola (green)

Projective equivalence to the cubic parabola

If one depicts Neil's unit parabola with the projective mapping (involutor perspective with the axis and center ), one obtains the curve , i.e. the cubic parabola . The tip (zero point) of Neil's parabola is exchanged for the far point of the y-axis. ${\displaystyle (t^{2},t^{3})}$ ${\displaystyle (x,y)\rightarrow ({\tfrac {x}{y}},{\tfrac {1}{y}})}$ ${\displaystyle y=1}$ ${\displaystyle (0,-1)}$ ${\displaystyle \left({\frac {1}{t}},{\frac {1}{t^{3}}}\right)}$ ${\displaystyle y=x^{3}}$

This property can also be seen in the representation of Neil's parabola in homogeneous coordinates : If one replaces in (A) (the line in the distance has the equation ) and multiplied by one obtains the curve equation ${\displaystyle x={\tfrac {x_{1}}{x_{3}}},\;y={\tfrac {x_{2}}{x_{3}}}}$ ${\displaystyle x_{3}=0}$ ${\displaystyle x_{3}^{3}}$

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

If one now chooses the straight line as the distance line and sets the (affine) curve ${\displaystyle x_{\color {red}2}=0}$ ${\displaystyle x={\tfrac {x_{1}}{x_{2}}},\;y={\tfrac {x_{3}}{x_{2}}}}$ ${\displaystyle y=x^{3}\;.}$

literature

• August Pein: The semicubic or Neil parabola, its secants and tangents , 1875, dissertation
• Clifford A. Pickover: The Length of Neile's Semicubical Parabola

Individual evidence

1. ^ Walter Gellert, Herbert Kästner , Siegfried Neuber (eds.): Lexicon of Mathematics , VEB Bibliographisches Institut Leipzig, 1979. S 461, rational curve .
2. ↑ August Pein: The semicubic or Neil parabola, their secants and tangents , p. 2
3. ^ Clifford A. Pickover: The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics , Sterling Publishing Company, 2009, ISBN 9781402757969 , p. 148
4. ↑ August Pein: The semicubic or Neil parabola, their secants and tangents , p. 26
5. ↑ August Pein: The semicubic or Neil parabola, their secants and tangents , p. 10