Geodesic
Definition and Calculation of an Eight-Centered Oval which is Quasi-Equivalent to the Ellipse
Journal for geometry and graphics, Tome 19 (2015) no. 2, pp. 257-268.

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Let Eb be an ellipse where b is the ratio of minor to major axis. We consider different approximations by ovals, which are composed from circular arcs and have also two axes of symmetry. We study (a) three four-centered ovals (quadrarcs), which share the vertices with the ellipse Eb. The first one has the same surface area, the second the minimum error of curvature at the vertices, and the third the same perimeter length. (b) Further, we investigate three eight-centered ovals, which also share the vertices with Eb. They have the same curvature at the vertices, and in addition, either the same surface area or the same perimeter length as Eb. As a conclusion, an eight-centered oval seems to be optimal and can therefore be called 'quasi-equivalent' to Eb. We show that in this case the difference of surface areas is rather small; the maximum is achieved at b = 0.1969. The maximum of the deformation error is achieved at b = 0.2379.
Classification : 51M04, 51N20
Mots-clés : Eight-centered oval, quadrarc, quasi-equivalent oval, ellipse 
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     author = {B. Herrera and A. Samper },
     title = {Definition and {Calculation} of an {Eight-Centered} {Oval} which is {Quasi-Equivalent} to the {Ellipse}},
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     pages = {257--268},
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B. Herrera; A. Samper . Definition and Calculation of an Eight-Centered Oval which is Quasi-Equivalent to the Ellipse. Journal for geometry and graphics, Tome 19 (2015) no. 2, pp. 257-268. http://geodesic.mathdoc.fr/item/JGG_2015_19_2_JGG_2015_19_2_a8/