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.
Author(s): Robert D. Poodiack , William E. Wood
Series: Springer Undergraduate Mathematics Series
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2022
Language: English
Pages: 289
Tags: Squigonometry, Generalized Trigonometry, p-Norm, Elliptic integrals, Generalized Exponentials
Preface
Audience and Structure
Acknowledgements
Rob’s Acknowledgements
Bill’s Acknowledgements
Contents
List of Projects
1 Introduction
2 Imperfection
3 A squigonometry introduction
3.1 Parameterizing the p-circle
3.2 Other squigonometric functions
3.3 Derivatives
3.4 A differential equation
4 p-metrics
4.1 Non-Euclidean geometry
4.2 Exploring p-metrics
4.3 Semimetrics
4.4 Conjugate metrics
4.5 Conics and other curves
5 Inverse squigonometric functions
5.1 Inverting sine and cosine
5.2 Inverting the rest
5.3 Composition of squig and inverse squig functions
6 The many values of Pi
6.1 Special functions
6.2 A simple definition
6.3 Exact values for the gamma function
6.4 Domains and ranges
7 Parameterizations
7.1 Interpreting the parameter
7.2 Areal trigonometric functions
7.3 Angular trigonometric functions
7.4 Inverse angular trigonometric functions
8 Arclength parameterization
8.1 p-Arclength
8.2 Pi redux
8.3 Arclength trigonometric functions
8.4 Inverse arclength trigonometric functions
8.5 Duality of area and arclength
9 Integrating squigonometric functions
9.1 Antiderivatives
9.2 Substitution
9.3 Powers of squigonometric functions
9.4 Squigonometric substitution
10 Three applications
10.1 The area of Sergels Torg
10.2 The volume of a superegg
10.3 The trisection of an area
11 Infinite series
11.1 Inverse squigonometric functions
11.2 Squigonometric functions
11.3 Convergence and complex numbers
12 Series and rational approximations
12.1 Series for Pi
12.2 Rational approximations to Pi
12.3 Generating new series and sums
13 Alternate coordinate systems
13.1 Squircular coordinate system
13.2 Squircular curves
13.3 Area
13.4 Double integrals
13.5 Orthogonal trajectories
13.6 Expanding to three dimensions
14 Hyperbolic functions
14.1 Definitions
14.2 Complex symmetry
14.3 The Gudermannian function
15 Exponentials and logarithms
15.1 Generalized exponentials
15.2 Series for exponential functions
15.3 Generalized logarithms
15.4 Logarithms and inverse hyperbolic functions
15.5 Series for logarithms
16 Elliptic integrals
16.1 Lemniscates
16.2 The lemniscate constant, Gauss and Pi
16.3 Addition formulas
16.4 Dixon and Weierstrass elliptic functions
17 More on lemniscates and ellipses
17.1 Lemniscatic functions
17.2 From elliptic to lemniscatic functions
17.3 Derivatives of lemniscatic functions
17.4 From lemniscatic to squigonometric functions
18 Geometry in the p-norm
18.1 Normed vector spaces
18.2 Convexity and Minkowski Geometry
19 Duality
19.1 The dual norm and pedal curves
19.2 Tangential parameterization
19.3 Hölder's inequality and friends
20 Analytic parameterizations
20.1 Generalizing the arcsine
20.2 Two other generalizations
20.3 Two-parameter functions
A Curve menagerie
B Formulas and integrals
B.1 Addition and doubling formulas
B.2 Integral table
B.3 Relationships between squigonometric functions and inverse squigonometric functions
B.4 Formulas for p=1/2
B.5 Gudermannian identities
C Parameterization primer
D Proofs of formulas and theorems
E Alternate Pi Days
F Selected exercise hints and solutions
References
Index
