Cardioids and Self-inversive Cubic Polynomials

Abstract

The standard cardioid is the set of points in the complex plane formed by reflecting the point 1 in every tangent to the unit circle. These points constitute a simple closed curve that is the boundary of two open disjoint regions, a bounded inner region that is heart-shaped, and an outer region. We verify that the outer region consists of all the points from which three tangents can be drawn to the cardioid, a statement that is part of the folklore of the theory of the cardioid—and deemed by many to be geometrically obvious! It was the key observation that led Frank Morley to his celebrated trisector theorem.

• MSC: Primary 30C10
• Secondary 14H45

Acknowledgment

The authors wish to thank Tom Carroll, who very kindly read an earlier draft of the paper, and the reviewers for their helpful comments.

Additional information

Notes on contributors

Finbarr Holland

FINBARR HOLLAND is professor emeritus at his alma mater University College Cork, where he graduated in 1962 with a master’s degree in mathematical science. He received his Ph.D. in harmonic analysis from the National University of Wales in 1964. He was elected the first president of the Irish Mathematical Society in 1977, and, eleven years later, led the first Irish team to participate in an International Mathematical Olympiad. He is a regular contributor to the problem section of several journals. [CrossRef]

Roger Smyth

ROGER SMYTH studied at Cambridge University from 1965 to 1969 and then became a research student of Trevor West at Trinity College Dublin. He received his Ph.D. in 1972. Thereafter he moved into Information Technology and worked in Queen’s University Belfast and the Northern Ireland Department of Health. His primary mathematical interest is Fredholm theory in Banach algebras, but he maintains a recreational interest in simple Euclidean geometry.