Rose Curves

Abstract

In this paper we study the rose curves in the polar coordinate system. Therefore, we first briefly describe their shape in the context of shapes in nature, architecture and also in cuisine. Then we describe the biography of Luigi Guido Grandi, who was the first to define the rose curve. We also introduce the polar coordinate system and explain how each point is related to the radial coordinate (or radius) and the angular coordinate (or polar angle), both called polar coordinates. We explain in detail the polar equation of a rose curve, where the radius is defined as the cosine function of the polar angle multiplied by a real constant called the angular frequency, which is considered here as a positive real number. In this way, we have considered the angular frequency as a positive integer (odd or even), as a positive rational number in the form of an irreducible fraction, or as a positive irrational number. We have explained and shown by some pictures that a rose curve is usually complete for any continuous interval subset of the set of real numbers, whose length depends on the angular frequency. In fact, a rose curve is incomplete only if the angular frequency is an irrational number. MSC Classification: 51A50 , 53A04 , 97G60 , 97I20