In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form
y2-a2x3=0
Solving for leads to the explicit form
y=\pma
| ||||
x |
,
Solving the implicit equation for yields a second explicit form
x=\left(
y | |
a |
| ||||
\right) |
.
x=t2, y=at3
The semicubical parabolas have a cuspidal singularity; hence the name of cuspidal cubic.
The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History).[2]
Any semicubical parabola
(t2,at3)
Proof: The similarity
(x,y) → (a2x,a2y)
(t2,at3)
((at)2,(at)3)=(u2,u3)
The parametric representation
(t2,at3)
(0,0)
t=0
Differentiating the semicubical unit parabola
y=\pmx3/2
(x0,y0)
y=
\sqrt{x0 | |
\left( | x0 |
4 |
,-
y0 | |
8 |
\right).
(x0,y0)
Determining the arclength of a curve
(x(t),y(t))
(t2,at3), 0\let\leb,
Example: For (semicubical unit parabola) and which means the length of the arc between the origin and point (4,8), one gets the arc length 9.073.
(t2,t)
In order to get the representation of the semicubical parabola
(t2,at3)
y=mx
m\ne0
m=\tan\varphi
\sec2\varphi=1+\tan2\varphi
r=\left( | \tan\varphi |
a |
\right)2\sec\varphi , -
\pi | |
2 |
<\varphi<
\pi | |
2 |
.
Mapping the semicubical parabola
(t2,t3)
y=1
y=x3.
This property can be derived, too, if one represents the semicubical parabola by homogeneous coordinates: In equation (A) the replacement
x=\tfrac{x1}{x3}, y=\tfrac{x2}{x3}
3 | |
x | |
3 |
x3
2 | |
x | |
2 |
-
3 | |
x | |
1 |
=0.
x\color{red2}=0
x=\tfrac{x1}{x2}, y=\tfrac{x3}{x2}
y=x3.
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.
The semicubical parabola was discovered in 1657 by William Neile who computed its arc length. Although the lengths of some other non-algebraic curves including the logarithmic spiral and cycloid had already been computed (that is, those curves had been rectified), the semicubical parabola was the first algebraic curve (excluding the line and circle) to be rectified.[1]