Squigonometry: The Study of Imperfect Circles

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.

“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)

• 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.