The quasi-squares and their limit curve

The main theme in the book "Introduction to Quasi-Quadrilaterals" is that of quasiquadrilateralsand their properties. The first topic is about quasi-square representedby the equation x^2n + y^2 = 1 or x^2 + y^2n = 1. It is well-defined: the curve liesbetween the unit circle and the specified square which has its center at the origin ofthe Cartesian system and, sides of length 2 which are parallel to the coordinative axes.These type of closed curves do not represent squares but for values of n larger than100 they are almost squares. From this phenomenon derives the name "quasisquare". Also, it is proved that the curve, represented by such equation, perfectly fitsto the sides of the specified square as n increases beyond bound. In this paper wepresent a more general case of the sequence of the quasi-squares and confirm theabove fitness by proving that there exists the limit curve of such a sequence. Othersubsidiary theorems are proved as well.