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VOL. 84, NO. 4, OCTOBER 2011

257

Squigonometry

W I L L I A M E. W O O D

University of Northern Iowa Cedar Falls, IA 50614 bill.wood@uni.edu

It is easy to take the circle for granted. In this paper, we look to enhance our appreciation of the circle by developing an analog of trigonometry—a subject built upon analysis of the circle—for something that is not quite a circle. Our primary model is the unit squircle, the superellipse deﬁned as the set of points (x, y) in the plane satisfying x 4 + y 4 = 1, depicted in F IGURE 1. It is a closed curve about the origin, but while any line through the origin is a line of symmetry of the circle, there are only four lines of symmetry for the squircle. Many familiar notions from trigonometry have natural analogs and we will see some interesting behaviors and results, but we will also see where the lower degree of symmetry inconveniences our new theory of squigonometry. We only scratch the surface here, offering many opportunities for the reader to extend the theory into studies of elliptic integrals, non-euclidean geometry, number theory, and complex analysis. 1 0.5 –1 –0.5 –0.5 –1 0.5 1

Figure 1 The unit squircle

Trigonometric functions and differential equations

We begin with the classical parameterization of the circle as the set of points of the form (cos t, sin t) for 0 ≤ t ≤ 2π . The cosine and sine functions report the x and y coordinates of the circle x 2 + y 2 = 1. There is nothing to stop us from doing the same thing for a squircle. We will deﬁne the functions as solutions to coupled initial value problems analogous to those that deﬁne the trigonometric functions. This approach is inspired by methods discussed in [1] and [2] of using IVP’s to develop transcendental functions in a ﬁrst-year calculus course. d d Recall from calculus that dt cos t = − sin t and dt sin t = cos t. We view these as deﬁning properties for this pair of functions. When we combine these relationships with initial conditions, we can say that cosine and sine are the functions satisfying x (t) = −y(t) y (t) = x(t) (1) x(0) = 1 y(0) = 0 where x corresponds to cosine and y to sine. This is an example of a coupled initial value problem, and it turns out that problems like this always have unique solutions. Math. Mag. 84 (2011) 257–265. doi:10.4169/math.mag.84.4.257. c Mathematical Association of America

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Therefore, we may deﬁne cosine and sine to be the unique solution to (1). We could then use Euler’s method to approximate values, and the graph of the phase plane would trace out the unit circle. Further, the equations (1) are enough to derive all of the familiar properties of these functions. For example, consider the function f (t) = u(t)2 + v(t)2 , whose derivative is given by f (t) = 2u(t)u (t) + 2v(t)v (t). If u(t) = cos(t) and v(t) = sin(t) are deﬁned by (1), then in this case f (t) = 0 and f (0) = 1. A function whose derivative is always zero must be constant, so f (t) = 1 for all t, or cos2 t + sin2 t = 1. This...