Studies On Oval Domains In The Euclidean Plane

In this paper, we investigate oval domains in the Euclidean plain IE2.Firstly, We study the radius of curvature of an oval domain (Theorem3.1, Theorem3.2) and the equation of the boundary of an oval domain in the Euclidean plain IE2(Theorem3.6, Theorem3.8), by the theory of ordinary differential equation of order2, the Fourier series of a periodic function and basic properties about oval domains in integral geometry.Secondly, we obtain the sufficient condition by which we can construct an oval domain by the radius of curvature (Theorem1.1). We obtain the equation of the isoperimetric deficit of an oval domain by the Fourier coefficients of its support function. We illustrate the geometrical significance of the deficit (Theorem4.3). Using the equations on isoperimetric deficit and the Steiner-point of an oval domain, we obtain a sufficient condition for an oval domain to be contained in another one by a rigid motion (Theorem5.2). We also obtain a necessary and sufficient condition for an oval domain to be of constant width and the sufficient condition to construct a constant width oval domain (Theorem6.1, Corollary6.2). We also obtain a necessary and sufficient condition for an oval domain to be original-symmetrical and a sufficient condition to construct an original-symmetrical oval domain (Theorem7.1, Corollary7.2) via the method of this paper.Finally, we obtain some geometric identities on the Pseudo-Reuleanx n-side domains defined by us. We check our results in the paper via programs of MATLAB.