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Journal for Geometry and Graphics 19 (2015), No. 2, 257--268
Copyright Heldermann Verlag 2015

Definition and Calculation of an Eight-Centered Oval which is Quasi-Equivalent to the Ellipse

Blas Herrera
Dep. d'Enginyeria InformÓtica i MatemÓtiques, Universitat Rovira i Virgili, Av. Pa´sos Catalans 26, 43007 Tarragona, Spain

Albert Samper
Unitat predepartamental d'Arquitectura, Universitat Rovira i Virgili, Av. Pa´sos Catalans 26, 43007 Tarragona, Spain

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.

Keywords: Eight-centered oval, quadrarc, quasi-equivalent oval, ellipse.

MSC: 51M04; 51N20

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