Given a Parabola
![\begin{displaymath}
y=x^2,
\end{displaymath}](p1_307.gif) |
(1) |
the parametric equation and its derivatives are
![\begin{displaymath}
\matrix{x=t\cr y=t^2\cr}\qquad
\matrix{x'=t\cr x''=0\cr}\qquad
\matrix{y'=2t\cr y''=2.\cr}
\end{displaymath}](p1_308.gif) |
(2) |
The Radius of Curvature is
![\begin{displaymath}
R={(x'^2+y'^2)^{3/2}\over x'y''-x''y'} = {(1+4t^2)^{3/2}\over 2}.
\end{displaymath}](p1_309.gif) |
(3) |
The Tangent Vector is
![\begin{displaymath}
\hat {\bf T} = {1\over\sqrt{1+4t^2}} \left[{\matrix{1\cr 2t\cr}}\right],
\end{displaymath}](p1_310.gif) |
(4) |
so the parametric equations of the evolute are
and
![\begin{displaymath}
{\textstyle{1\over 3}}(\eta-{\textstyle{1\over 2}})=(-{\textstyle{1\over 4}}\xi)^{2/3}
\end{displaymath}](p1_319.gif) |
(9) |
![\begin{displaymath}
{\textstyle{1\over 3}}(\eta-h) = \left({-{2\xi\over 8}}\right)^{2/3}= {\textstyle{1\over 4}}(2\xi)^{2/3}.
\end{displaymath}](p1_320.gif) |
(10) |
The Evolute is therefore
![\begin{displaymath}
\eta={\textstyle{3\over 4}} (2\xi)^{2/3}+{\textstyle{1\over 2}}.
\end{displaymath}](p1_321.gif) |
(11) |
This is known as Neile's Parabola and is a Semicubical Parabola. From a point above the evolute three
normals can be drawn to the Parabola, while only one normal can be drawn to the Parabola from a point
below the Evolute.
See also Neile's Parabola, Parabola, Semicubical Parabola
© 1996-9 Eric W. Weisstein
1999-05-26