This textbook introduces generalized trigonometric functions through the exploration of imperfect circles: curves defined by |x|p + |y|p = 1 where p ≥ 1. Grounded in visualization and computations, this accessible, modern perspective encompasses new and old results, casting a fresh light on duality, special functions, geometric curves, and differential equations. Projects and opportunities for research abound, as we explore how similar (or different) the trigonometric and squigonometric worlds might be.
Comprised of many short chapters, the book begins with core definitions and techniques. Successive chapters cover inverse squigonometric functions, the many possible re-interpretations of π, two deeper dives into parameterizing the squigonometric functions, and integration. Applications include a celebration of Piet Hein’s work in design. From here, more technical pathways offer further exploration. Topicsinclude infinite series; hyperbolic, exponential, and logarithmic functions; metrics and norms; and lemniscatic and elliptic functions. Illuminating illustrations accompany the text throughout, along with historical anecdotes, engaging exercises, and wry humor.
Squigonometry: The Study of Imperfect Circles invites readers to extend familiar notions from trigonometry into a new setting. Ideal for an undergraduate reading course in mathematics or a senior capstone, this book offers scaffolding for active discovery. Knowledge of the trigonometric functions, single-variable calculus, and initial-value problems is assumed, while familiarity with multivariable calculus and linear algebra will allow additional insights into certain later material.
Table of Contents:
1. Introduction.- 2. Imperfection.- 3. A Squigonometry Introduction.- 4. p-metrics.- 5. Inverse squigonometric functions.- 6. The many values of Pi.- 7. Parametrizations.- 8. Arclength Parametrization.- 9. Integrating Squigonometric Functions.- 10. Three applications.- 11. Infinite series.- 12. Series and rational approximations.- 13. Alternate Coordinates.- 14. Hyperbolic Functions.- 15. Exponentials and Logarithms.- 16. Elliptic Integrals.- 17. Lemniscates and Ellipses.- 18. Geometry in the p-norm.- 19. Duality.- 20. Analytic Parametrizations.- A. Curve Menagerie.- B. Formulas and Integrals.- C. Parametrization Primer.- D. Proofs of Formulas and Theorems.- E. Alternate Pi Days.- F. Selected Exercise Hints and Solutions.
About the Author :
Robert D. Poodiack is Professor of Mathematics at Norwich University, Vermont, the nation's oldest private military college. He earned his PhD in mathematical sciences from the University of Vermont in 1999. He has supervised dozens of senior capstone projects in his 23 years at Norwich, many on squigonometry-related topics. He is very active in the Northeastern Section of the Mathematical Association of America. In his spare time, he enjoys running, baking, and cooking.
William E. Wood is an Associate Professor of Mathematics at the University of Northern Iowa. His mathematical pursuits tend toward geometry, including discrete conformal geometry, visualization, applications, and analysis, with particular interest in finding ways to bring geometric ideas to students in the classroom and through research projects. He earned his PhD from Florida State University.
Review :
"I highly recommend this book mainly because it is not a run-of-the-mill textbook; instead it discusses a host of beautiful problems from geometry and analysis and introduces power series, Fourier series, differential equations, continued fractions, parametrizations of curves, elliptic integrals and lemniscatic functions, and a lot more. Of course there are exercises, and index, and an extensive bibliography." (Franz Lemmermeyer, zbMATH 1560.33001, 2025)
"In this monograph, the authors have compiled results about variants of sine and cosine functions ... . the notions and results needed to resolve them can be found in the text. ... The book will therefore be suitable both in the context of an undergraduate mathematics course with directed reading and for end-of-undergraduate study research projects. Incidentally, 25 descriptions of projects intended to serve as a springboard for independent study are included throughout the book." (Frédéric Morneau-Guérin, MAA Reviews, August 27, 2023)
