
Journal for Geometry and Graphics 19 (2015), No. 2, 257268 Copyright Heldermann Verlag 2015 Definition and Calculation of an EightCentered Oval which is QuasiEquivalent to the Ellipse Blas Herrera Dep. d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain blas.herrera@urv.net Albert Samper Unitat predepartamental d'Arquitectura, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Spain albert.samper@urv.net Let E_{b} 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 fourcentered ovals (quadrarcs), which share the vertices with the ellipse E_{b}. 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 eightcentered ovals, which also share the vertices with E_{b}. They have the same curvature at the vertices, and in addition, either the same surface area or the same perimeter length as E_{b}. As a conclusion, an eightcentered oval seems to be optimal and can therefore be called 'quasiequivalent' to E_{b}. 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: Eightcentered oval, quadrarc, quasiequivalent oval, ellipse. MSC: 51M04; 51N20 [ Fulltextpdf (233 KB)] for subscribers only. 