Editing Ellipse (section)

=== Curvature ===
The [[curvature]] is given by:

<math display="block">\kappa = \frac{1}{a^2 b^2}\left(\frac{x^2}{a^4}+\frac{y^2}{b^4}\right)^{-\frac{3}{2}}\ ,</math>

and the [[Radius of curvature#Ellipses|radius of curvature]], ρ = 1/κ, at point <math>(x,y)</math>:
<math display="block">\rho = a^2 b^2 \left(\frac{x^{2}}{a^4} + \frac{y^{2}}{b^4}\right)^\frac{3}{2} = \frac{1}{a^4 b^4} \sqrt{\left(a^4 y^{2} + b^4 x^{2}\right)^3} \ .</math>The radius of curvature of an ellipse, as a function of angle ''{{mvar|θ}}'' from the center, is:
<math display="block">R(\theta)=\frac{a^2}{b}\biggl(\frac{1-e^2(2-e^2)(\cos\theta)^2)}{1-e^2(\cos\theta)^2}\biggr)^{3/2}\,,</math>where e is the eccentricity.

Radius of curvature at the two ''vertices'' <math>(\pm a,0)</math> and the centers of curvature:
<math display="block">\rho_0 = \frac{b^2}{a}=p\ , \qquad \left(\pm\frac{c^2}{a}\,\bigg|\,0\right)\ .</math>

Radius of curvature at the two ''co-vertices'' <math>(0,\pm b)</math> and the centers of curvature:
<math display="block">\rho_1 = \frac{a^2}{b}\ , \qquad \left(0\,\bigg|\,\pm\frac{c^2}{b}\right)\ .</math>The locus of all the centers of curvature is called an [[Evolute#Evolute of an ellipse|evolute]]. In the case of an ellipse, the evolute is an [[astroid]].