On the Isoptic Hypersurfaces in the n-Dimensional Euclidean Space

The theory of the isoptic curves is widely studied in the Euclidean plane E2 (see [1] and [13] and the references given there). The analogous question was investigated by the authors in the hyperbolic H2 and elliptic E2 planes (see [3], [4]), but in the higher dimensional spaces there is no result according to this topic. In this paper we give a natural extension of the notion of the isoptic curves to the n-dimensional Euclidean space En (n ≥ 3) which are called isoptic hypersurfaces. We develope an algorithm to determine the isoptic hypersurface HD of an arbitrary (n−1) dimensional compact parametric domain D lying in a hyperplane in the Euclidean n-space. We will determine the equation of the isoptic hypersurfaces of rectangles D ⊂ E2 and visualize them with Wolfram Mathematica. Moreover, we will show some possible applications of the isoptic hypersurfaces.