Ellipse
Curvature

The curvature is given by:

κ = 1 a 2 b 2 (x 2 a 4 + y 2 b 4) − 3 2 , {\displaystyle \kappa ={\frac {1}{a^{2}b^{2}}}\left({\frac {x^{2}}{a^{4}}}+{\frac {y^{2}}{b^{4}}}\right)^{-{\frac {3}{2}}}\,}

and the radius of curvature, ρ = 1/κ, at point (x, y) {\displaystyle (x,y)}: ρ = a 2 b 2 (x 2 a 4 + y 2 b 4) 3 2 = 1 a 4 b 4 (a 4 y 2 + b 4 x 2) 3. {\displaystyle \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}}}\.} The radius of curvature of an ellipse, as a function of angle θ from the center, is: R (θ) = a 2 b (1 − e 2 (2 − e 2) (cos θ) 2) 1 − e 2 (cos θ) 2) 3 / 2, {\displaystyle 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}\,,} where e is the eccentricity.

Radius of curvature at the two vertices (± a, 0) {\displaystyle (\pm a,0)} and the centers of curvature: ρ 0 = b 2 a = p , (± c 2 a | 0). {\displaystyle \rho _{0}={\frac {b^{2}}{a}}=p\,\qquad \left(\pm {\frac {c^{2}}{a}}\,{\bigg |}\,0\right)\.}

Radius of curvature at the two co-vertices (0, ± b) {\displaystyle (0,\pm b)} and the centers of curvature: ρ 1 = a 2 b , (0 | ± c 2 b). {\displaystyle \rho _{1}={\frac {a^{2}}{b}}\,\qquad \left(0\,{\bigg |}\,\pm {\frac {c^{2}}{b}}\right)\.} The locus of all the centers of curvature is called an evolute. In the case of an ellipse, the evolute is an astroid.
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