squigonometry

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.{{cite book |last=Holton |first=Derek |year=1990 |title=Creating Problems: Counting; Packing; Intersecting; Chessboards; Squigonometry |series=Derek Holton's problem solving series |volume=15 |publisher = University of Otago |isbn=0-908903-15-4}} Reprised in {{cite book |last=Holton |first=Derek |title=A Second Step to Mathematical Olympiad Problems |chapter=Squigonometry |at=§{{nbsp}}7.6, {{pgs|233–235}} |year=2011 |doi=10.1142/7979 |isbn=978-981-4327-87-9 |location=Singapore |publisher=World Scientific }} 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.{{Cite book |last=Poodiack |first=Robert D. |last2=Wood |first2=William E. |title=Squigonometry: The Study of Imperfect Circles |publisher=Springer |year=2022 |doi=10.1007/978-3-031-13783-9 |isbn=978-3-031-13782-2 |pages=1}} {{pb}}

Examples: {{pb}}

{{cite book |last=Lundberg |first=E. |year=1879 |title=Om hypergoniometriska funktioner af komplexa variabla |type=Manuscript }} Translation by Jaak Peetre (2000) [https://web.archive.org/web/20161024183030/http://www.maths.lth.se/matematiklu/personal/jaak/hypergf.ps "On hypergoniometric functions of complex variables"] (Postscript file). {{pb}}

{{cite journal |last=Shelupsky |first=D. |year=1959 |title=A generalization of the trigonometric functions |journal=The American Mathematical Monthly |volume=66 |number=10 |pages=879–884 |jstor=2309789 }}

File:Unit-squircle-graph-p-1-2-3-4.png

The cosquine and squine functions, denoted as $\backslash operatorname\{cq\}\_p(t)$ and $\backslash operatorname\{sq\}\_p(t),$ 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:

:$|x|^p\; +\; |y|^p\; =\; 1$

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

Notably, when $p=2$, 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 determined{{Cite journal |last=Elbert |first=Á. |date=1987-09-01 |title=On the half-linear second order differential equations |url=https://doi.org/10.1007/BF01951012 |journal=Acta Mathematica Hungarica |language=en |volume=49 |issue=3 |pages=487–508 |doi=10.1007/BF01951012 |issn=1588-2632|url-access=subscription }} by solving the coupled initial value problem{{cite journal |last1=Wood |first1=William E.|date=October 2011 |title=Squigonometry|url=https://doi.org/10.4169/math.mag.84.4.257|journal=Mathematics Magazine |volume=84 |issue=4 |pages=264|doi= |access-date=}}{{cite journal |last1=Chebolu |first1=Sunil|last2=Hatfield|first2=Andrew|last3=Klette|first3=Riley|last4=Moore|first4=Cristopher|last5=Warden|first5=Elizabeth|date=Fall 2022|title=Trigonometric functions in the p-norm|url=https://digitalresearch.bsu.edu/mathexchange/volume-16/|pages=4, 5|journal=BSU Undergraduate Mathematics Exchange|volume=16|issue=1|doi= |access-date=}}

:$\backslash begin\{cases\}$

x'(t)=-|y(t)|^{p-1}\\

y'(t)=|x(t)|^{p-1}\\

x(0)=1\\

y(0)=0

\end{cases}

Where $x$ corresponds to $\backslash operatorname\{cq\}\_p(t)$ and $y$ corresponds to $\backslash operatorname\{sq\}\_p(t)$.{{cite book |last1=Girg |first1=Petr E.|last2=Kotrla|first2=Lukáš|date=February 2014|title=Differentiability properties of p-trigonometric functions|url=https://www.researchgate.net/publication/262335988_Differentiability_properties_of_p-trigonometric_functions |pages=104|doi= |access-date=}}

The definition of sine and cosine through integrals can be extended to define the squigonometric functions. Let

:$F\_p\; (x)=\backslash int\_\{0\}^\{x\}\backslash frac\{1\}\{\{(1-t^p)\}^\backslash tfrac\{p-1\}\{p\}\}\backslash ,dt$

Since $F\_p$ is strictly increasing it is a one-to-one function on $[0,1]$ with range $[0,\backslash pi\_p/2]$, where $\backslash pi\_p$ is defined as follows:

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

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

:$\backslash operatorname\{sq\}\_p\; (x)=\backslash operatorname\{sq\}\_p\; (\backslash pi\_p-x)$

By this means $sq\_p$ is differentiable in $\{\{\backslash R\}\}$ and, corresponding to this, the function $cq\_p$ is defined by:

:$\backslash frac\{d\}\{dx\}\backslash operatorname\{sq\}\_p\; (x)\; =\; \backslash operatorname\{cq\}\_p(x)^\{p-1\}.$

The tanquent, cotanquent, sequent and cosequent functions can be defined as follows:{{r|SHS}}{{cite journal |last1=Edmunds |first1=David E. |last2=Gurka |first2=Petr|last3=Lang|first3=Jan|date=2012 |title=Properties of generalized trigonometric functions | url= https://doi.org/10.1016/j.jat.2011.09.004|journal=Journal of Approximation Theory|volume=164 |issue=1 |pages=49 |doi= |access-date=}}

:$\backslash operatorname\{tq\}\_p(t)=\backslash frac\{\backslash operatorname\{sq\}\_p(t)\}\{\backslash operatorname\{cq\}\_p(t)\}$

:$\backslash operatorname\{ctq\}\_p(t)=\backslash frac\{\backslash operatorname\{cq\}\_p(t)\}\{\backslash operatorname\{sq\}\_p(t)\}$

:$\backslash operatorname\{seq\}\_p(t)=\backslash frac\{1\}\{\backslash operatorname\{cq\}\_p(t)\}$

:$\backslash operatorname\{cseq\}\_p(t)=\backslash frac\{1\}\{\backslash operatorname\{sq\}\_p(t)\}$

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

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

Similarly, the inverse squine is defined as:

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

Other parameterizations of squircles give rise to alternate definitions of these functions. For example, Edmunds, Lang, and Gurka {{Cite book |last=Edmunds |first=David |title=Eigenvalues, Embeddings and Generalised Trigonometric Functions |publisher=Springer-Verlag Berlin Heidelberg |year=2011}} define $\backslash tilde\; F\_\; p(x)$ as:

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

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

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

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

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

Thus $\backslash cos\_p$ is strictly decreasing on $[0,\backslash tilde\; \backslash pi\_p/2]$, $\backslash cos\_p(0)=1$ and $\backslash cos\_p(\backslash tilde\; \backslash pi\_2/2)=0$. Also:

$|\backslash sin\_px|^p+|\backslash cos\_px|^p=1$ .

This is immediate if $x\; \backslash in\; [0,\backslash tilde\; \backslash pi/2\; ]$, but it holds for all $x\; \backslash in\; \backslash R$ 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 form{{cite journal |last1=Poodiack |first1=Robert D. |date=April 2016 |title=Squigonometry, Hyperellipses, and Supereggs. |url=https://doi.org/10.4169/math.mag.89.2.92 |journal=Mathematics Magazine |volume=89 |issue=2 |pages=92-102 |doi=10.4169/math.mag.89.2.92 |url-access=subscription }}

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

that are otherwise computationally difficult to handle.

Squigonometry has been applied to find expressions for the volume of superellipsoids, such as the superegg.{{r|SHS}}

• Astroid
• Ellipsoid
• Lp space
• Oval
• Squround

{{reflist}}

Category:Trigonometry