Agnes Scott College

Sister Mary Gervase

On the cardioids fulfilling certain assigned conditions
Catholic University of America, 1917

Introduction

The tricuspidal, bicircular quartic of the third class defined by the

Cartesian equation (x2+y2+ax)2=a2(x2+y2)
polar equation ρ=a(1-cos θ)

and commonly known as the Cardioid, has for many years been the object of mathematical investigation. It has lately been studied by Raymond Clare Archibald in his Inaugural-Dissertation "The Cardioide and Some of Its Related Curves" (Strassburg, 1900), which work contains an historical sketch of the curve and a presentation of results prior to the year of its publication. Since then, the only work on the subject of considerable length is Professor Archibald's paper, "The Cardioid and Tricuspid: Quartics with Three Cusps.". Besides this there have appeared a few detached problems and contributions in periodicals treating the curve from either a metric or a projective standpoint.

The chief characteristic of former research along this line seems to be the examination of the cardioid as a fixed curve and the consideration of its properties as such. The present investigation starts from a different point of view, which we may outline as follows:

in general, a curve of the fourth degree is capable of satisfying 14 conditions. The cardioid, however, having 3 cusps, two of which are at the fixed (circular) points I and J, can be subjected to only 4 conditions. If, then, 3 conditions be imposed, there are ∞1 curves satisfying them; therefore, the special elements (cusp, focus, double tangent) describe definite loci. If 4 conditions are given, there are a finite number of curves satisfying them. It is our purpose to obtain the loci generated in the first case; and, in the second, to determine the number and (where possible) the reality of cardioids for various kinds of assigned conditions.