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poodiack r squigonometry study imperfect circles 2022

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Textbook in PDF format 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. 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. Audience and Structure Acknowledgements Rob’s Acknowledgements Bill’s Acknowledgements List of Projects Imperfection A squigonometry introduction Parameterizing the p-circle Other squigonometric functions Derivatives A differential equation p-metrics Non-Euclidean geometry Exploring p-metrics Semimetrics Conjugate metrics Conics and other curves Inverse squigonometric functions Inverting sine and cosine Inverting the rest Composition of squig and inverse squig functions The many values of Pi Special functions A simple definition Exact values for the gamma function Domains and ranges Parameterizations Interpreting the parameter Areal trigonometric functions Angular trigonometric functions Inverse angular trigonometric functions Arclength parameterization p-Arclength Pi redux Arclength trigonometric functions Inverse arclength trigonometric functions Duality of area and arclength Integrating squigonometric functions Antiderivatives Substitution Powers of squigonometric functions Squigonometric substitution Three applications The area of Sergels Torg The volume of a superegg The trisection of an area Infinite series Inverse squigonometric functions Squigonometric functions Convergence and complex numbers Series and rational approximations Series for Pi Rational approximations to Pi Generating new series and sums Alternate coordinate systems Squircular coordinate system Squircular curves Area Double integrals Orthogonal trajectories Expanding to three dimensions Hyperbolic functions Definitions Complex symmetry The Gudermannian function Exponentials and logarithms Generalized exponentials Series for exponential functions Generalized logarithms Logarithms and inverse hyperbolic functions Series for logarithms Elliptic integrals Lemniscates The lemniscate constant, Gauss and Pi Addition formulas Dixon and Weierstrass elliptic functions More on lemniscates and ellipses Lemniscatic functions From elliptic to lemniscatic functions Derivatives of lemniscatic functions From lemniscatic to squigonometric functions Geometry in the p-norm Normed vector spaces Convexity and Minkowski Geometry Duality The dual norm and pedal curves Tangential parameterization Hölder's inequality and friends Analytic parameterizations Generalizing the arcsine Two other generalizations Two-parameter functions Curve menagerie Formulas and integrals Addition and doubling formulas Integral table Relationships between squigonometric functions and inverse squigonometric functions Formulas for p=1/2 Gudermannian identities Parameterization primer Proofs of formulas and theorems Alternate Pi Days Selected exercise hints and solutions

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poodiack r squigonometry study imperfect circles 2022