Squigonometry

Squigonometry or p-trigonometry is a generalization of traditional trigonometry which replaces the circle and Euclidean distance function with the squircle (shape intermediate between a square and circle) and p-norm. While trigonometry deals with the relationships between angles and lengths in the plane using trigonometric functions defined relative to a unit circle, squigonometry focuses on analogous relationships and functions within the context of a unit squircle.

The term squigonometry is a portmanteau of square or squircle and trigonometry. It was used by Derek Holton to refer to an analog of trigonometry using a square as a basic shape (instead of a circle) in his 1990 pamphlet Creating Problems.¹ In 2011 it was used by William Wood to refer to trigonometry with a squircle as its base shape in a recreational mathematics article in Mathematics Magazine. In 2016 Robert Poodiack extended Wood's work in another Mathematics Magazine article. Wood and Poodiack published a book about the topic in 2022.

However, the idea of generalizing trigonometry to curves other than circles is centuries older.²

The cosquine and squine functions, denoted as ${\displaystyle \operatorname {cq} _{p}(t)}$ and ${\displaystyle \operatorname {sq} _{p}(t),}$ can be defined analogously to trigonometric functions on a unit circle, but instead using the coordinates of points on a unit squircle, described by the equation:

${\displaystyle |x|^{p}+|y|^{p}=1}$

where ${\displaystyle p}$ is a real number greater than or equal to 1. Here ${\displaystyle x}$ corresponds to ${\displaystyle \operatorname {cq} _{p}(t)}$ and ${\displaystyle y}$ corresponds to ${\displaystyle \operatorname {sq} _{p}(t)}$

Notably, when ${\displaystyle p=2}$, the squigonometric functions coincide with the trigonometric functions.

Similarly to how trigonometric functions are defined through differential equations, the cosquine and squine functions are also uniquely determined³ by solving the coupled initial value problem⁴⁵

${\displaystyle {\begin{cases}x'(t)=-|y(t)|^{p-1}\\y'(t)=|x(t)|^{p-1}\\x(0)=1\\y(0)=0\end{cases}}}$

Where ${\displaystyle x}$ corresponds to ${\displaystyle \operatorname {cq} _{p}(t)}$ and ${\displaystyle y}$ corresponds to ${\displaystyle \operatorname {sq} _{p}(t)}$.⁶

The definition of sine and cosine through integrals can be extended to define the squigonometric functions. Let ${\displaystyle 1<p<\infty }$ and define a differentiable function ${\displaystyle F_{p}:[0,1]\rightarrow {\mathbb {R} }}$ by:

${\displaystyle F_{p}(x)=\int _{0}^{x}{\frac {1}{{(1-t^{p})}^{\tfrac {p-1}{p}}}}\,dt}$

Since ${\displaystyle F_{p}}$ is strictly increasing it is a one-to-one function on ${\displaystyle [0,1]}$ with range ${\displaystyle [0,\pi _{p}/2]}$, where ${\displaystyle \pi _{p}}$ is defined as follows:

${\displaystyle \pi _{p}=2\int _{0}^{1}{\frac {1}{{(1-t^{p})}^{\tfrac {p-1}{p}}}}\,dt}$

Let ${\displaystyle \operatorname {sq} _{p}}$ be the inverse of ${\displaystyle F_{p}}$ on ${\displaystyle [0,\pi _{p}/2]}$. This function can be extended to ${\displaystyle [0,\pi _{p}]}$ by defining the following relationship:

${\displaystyle \operatorname {sq} _{p}(x)=\operatorname {sq} _{p}(\pi _{p}-x)}$

By this means ${\displaystyle \operatorname {sq} _{p}}$ is differentiable in ${\displaystyle {\mathbb {R} }}$ and, corresponding to this, the function ${\displaystyle \operatorname {cq} _{p}}$ is defined by:

${\displaystyle {\frac {d}{dx}}\operatorname {sq} _{p}(x)=\operatorname {cq} _{p}(x)^{p-1}.}$

The tanquent, cotanquent, sequent and cosequent functions can be defined as follows:⁷⁸

${\displaystyle \operatorname {tq} _{p}(t)={\frac {\operatorname {sq} _{p}(t)}{\operatorname {cq} _{p}(t)}}}$

${\displaystyle \operatorname {ctq} _{p}(t)={\frac {\operatorname {cq} _{p}(t)}{\operatorname {sq} _{p}(t)}}}$

${\displaystyle \operatorname {seq} _{p}(t)={\frac {1}{\operatorname {cq} _{p}(t)}}}$

${\displaystyle \operatorname {csq} _{p}(t)={\frac {1}{\operatorname {sq} _{p}(t)}}}$

General versions of the inverse squine and cosquine can be derived from the initial value problem above. Let ${\displaystyle x=\operatorname {cq} _{p}(y)}$; by the inverse function rule, ${\displaystyle {\frac {dx}{dy}}=-[\operatorname {sq} _{p}(y)]^{p-1}=(1-x^{p})^{(p-1)/p}}$. Solving for ${\displaystyle y}$ gives the definition of the inverse cosquine:

${\displaystyle y=\operatorname {cq} _{p}^{-1}(x)=\int _{x}^{1}{\frac {1}{(1-t^{p})^{\frac {p-1}{p}}}}\,dt}$

Similarly, the inverse squine is defined as:

${\displaystyle \operatorname {sq} _{p}^{-1}(x)=\int _{0}^{x}{\frac {1}{(1-t^{p})^{\frac {p-1}{p}}}}\,dt}$

Other parameterizations of squircles give rise to alternate definitions of these functions. For example, Edmunds, Lang, and Gurka ⁹ define ${\displaystyle {\tilde {F}}_{p}(x)}$ as:

${\displaystyle {\tilde {F}}_{p}(x)=\int _{0}^{x}(1-t^{p})^{-(1/p)}\,dt}$.

Since ${\displaystyle F_{p}}$ is strictly increasing it has a =n inverse which, by analogy with the case ${\displaystyle p=2}$, we denote by ${\displaystyle \sin _{p}}$. This is defined on the interval ${\displaystyle [0,\pi _{p}/2]}$, where ${\displaystyle {\tilde {\pi }}_{p}}$ is defined as follows:

${\displaystyle {\tilde {\pi }}_{p}=2\int _{0}^{1}(1-t^{p})^{-(1/p)}\,dt}$.

Because of this, we know that ${\displaystyle \sin _{p}}$ is strictly increasing on ${\displaystyle [0,{\tilde {\pi }}_{p}/2]}$, ${\displaystyle \sin _{p}(0)=0}$ and ${\displaystyle \sin _{p}({\tilde {\pi }}_{p}/2)=1}$. We extend ${\displaystyle \sin _{p}}$ to ${\displaystyle [0,{\tilde {\pi }}_{p}]}$ by defining:

${\displaystyle \sin _{p}(x)=\sin _{p}({\tilde {\pi }}_{p}-x)}$ for ${\displaystyle x\in [{\tilde {\pi }}_{p}/2,{\tilde {\pi }}_{p}]}$ Similarly ${\displaystyle \cos _{p}(x)=(1-(\sin _{p}(x))^{p})^{\frac {1}{p}}}$.

Thus ${\displaystyle \cos _{p}}$ is strictly decreasing on ${\displaystyle [0,{\tilde {\pi }}_{p}/2]}$, ${\displaystyle \cos _{p}(0)=1}$ and ${\displaystyle \cos _{p}({\tilde {\pi }}_{2}/2)=0}$. Also:

${\displaystyle |\sin _{p}x|^{p}+|\cos _{p}x|^{p}=1}$ .

This is immediate if ${\displaystyle x\in [0,{\tilde {\pi }}/2]}$, but it holds for all ${\displaystyle x\in \mathbb {R} }$ in view of symmetry and periodicity.

Squigonometric substitution can be used to solve indefinite integrals using a method akin to trigonometric substitution, such as integrals in the generic form⁷

${\displaystyle I=\int ({1-t^{p}})^{\frac {1}{p}}\,dt}$

that are otherwise computationally difficult to handle.

Squigonometry has been applied to find expressions for the volume of superellipsoids, such as the superegg.⁷