Ellipse Involute
EllipseInvolute
The involute of an ellipse specified parametrically by
x = acost
(1)
y = bsint
(2)
is given by the parametric equations
![画像:a[cost+(bE(t,e)sint)/(sqrt(a^2sin^2t+b^2cos^2t))]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/EllipseInvolute/Inline9.svg){kind=link}
![画像:b[sint+(bE(t,e)cost)/(sqrt(a^2sin^2t+b^2cos^2t))],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/EllipseInvolute/Inline12.svg){kind=link}
where E(x,k) is an elliptic integral of the second kind, and
| [画像: e=sqrt(1-(a^2)/(b^2)). ] |
(5)
|
is the eccentricity.
The curvature and tangential angle are given by
kappa(t) = [画像:1/(bE(t,e))]
(6)
phi(t) = [画像:tan^(-1)((atant)/b),]
(7)
where E(x,k) is an elliptic integral of the second kind.
See also
Ellipse, Ellipse Evolute, InvoluteExplore with Wolfram|Alpha
WolframAlpha
More things to try:
Cite this as:
Weisstein, Eric W. "Ellipse Involute." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipseInvolute.html