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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 pi, 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. Topics include 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.

Squigonometry: The Study of Imperfect Circles

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.