Squigonometry

{{Short description|Branch of mathematics}}

'''Squigonometry''' or '''{{math|''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 {{math|''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'''.

== Etymology == 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''.<ref>{{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 }}</ref> 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.<ref>{{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 }} </ref>

== Squigonometric functions ==

===Cosquine and squine=== ====Definition through unit squircle==== thumb|Unit squircle for different values of p The cosquine and squine functions, denoted as <math>\operatorname{cq}_p(t)</math> and <math>\operatorname{sq}_p(t),</math> 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: :<math>|x|^p + |y|^p = 1</math> where <math>p</math> is a real number greater than or equal to 1. Here <math>x</math> corresponds to <math>\operatorname{cq}_p(t)</math> and <math>y</math> corresponds to <math>\operatorname{sq}_p(t)</math>

Notably, when <math>p=2</math>, the squigonometric functions coincide with the trigonometric functions.

====Definition through differential equations==== Similarly to how trigonometric functions are defined through differential equations, the cosquine and squine functions are also uniquely determined<ref>{{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 }}</ref> by solving the coupled initial value problem<ref>{{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=}}</ref><ref>{{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=}}</ref> :<math>\begin{cases} x'(t)=-|y(t)|^{p-1}\\ y'(t)=|x(t)|^{p-1}\\ x(0)=1\\ y(0)=0 \end{cases}</math> Where <math>x</math> corresponds to <math>\operatorname{cq}_p(t)</math> and <math>y</math> corresponds to <math>\operatorname{sq}_p(t)</math>.<ref>{{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=}}</ref>

====Definition through analysis==== The definition of sine and cosine through integrals can be extended to define the squigonometric functions. Let <math>1<p<\infty</math> and define a differentiable function <math>F_p:[0,1]\rightarrow{{\R}}</math> by: :<math>F_p (x)=\int_{0}^{x}\frac{1}{{(1-t^p)}^\tfrac{p-1}{p}}\,dt</math> Since <math>F_p</math> is strictly increasing it is a one-to-one function on <math>[0,1]</math> with range <math>[0,\pi_p/2]</math>, where <math>\pi_p</math> is defined as follows: :<math>\pi_p=2\int_{0}^{1}\frac{1}{{(1-t^p)}^\tfrac{p-1}{p}}\,dt</math> Let <math>\operatorname{sq}_p</math> be the inverse of <math>F_p</math> on <math>[0,\pi_p/2]</math>. This function can be extended to <math>[0,\pi_p]</math> by defining the following relationship: :<math>\operatorname{sq}_p (x)=\operatorname{sq}_p (\pi_p-x)</math> By this means <math>\operatorname{sq}_p</math> is differentiable in <math>{{\R}}</math> and, corresponding to this, the function <math>\operatorname{cq}_p</math> is defined by: :<math>\frac{d}{dx}\operatorname{sq}_p (x) = \operatorname{cq}_p(x)^{p-1}.</math>

===Tanquent, cotanquent, sequent and cosequent=== The tanquent, cotanquent, sequent and cosequent functions can be defined as follows:{{r|SHS}}<ref>{{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=}}</ref> :<math>\operatorname{tq}_p(t)=\frac{\operatorname{sq}_p(t)}{\operatorname{cq}_p(t)}</math> :<math>\operatorname{ctq}_p(t)=\frac{\operatorname{cq}_p(t)}{\operatorname{sq}_p(t)}</math> :<math>\operatorname{seq}_p(t)=\frac{1}{\operatorname{cq}_p(t)}</math> :<math>\operatorname{csq}_p(t)=\frac{1}{\operatorname{sq}_p(t)} </math>

===Inverse squigonometric functions=== General versions of the inverse squine and cosquine can be derived from the initial value problem above. Let <math>x=\operatorname{cq}_p (y)</math>; by the inverse function rule, <math>\frac{dx}{dy} =-[\operatorname{sq}_p (y)]^{p-1}=(1-x^p)^{(p-1)/p} </math>. Solving for <math>y</math> gives the definition of the inverse cosquine: :<math>y=\operatorname{cq}_{p}^{-1}(x) = \int_{x}^{1}\frac{1}{(1-t^p)^{\frac{p-1}{p}}}\,dt</math> Similarly, the inverse squine is defined as: :<math>\operatorname{sq}_{p}^{-1}(x) = \int_{0}^{x}\frac{1}{(1-t^p)^{\frac{p-1}{p}}}\,dt</math>

=== Multiple ways to approach squigonometry === Other parameterizations of squircles give rise to alternate definitions of these functions. For example, Edmunds, Lang, and Gurka <ref>{{Cite book |last=Edmunds |first=David |title=Eigenvalues, Embeddings and Generalised Trigonometric Functions |publisher=Springer-Verlag Berlin Heidelberg |year=2011}}</ref> define <math>\tilde F_ p(x)</math> as:

<math>\tilde F_p (x)= \int_{0}^{x}(1-t^p)^{-(1/p)}\,dt</math>.

Since <math>F_p</math> is strictly increasing it has a =n inverse which, by analogy with the case <math>p=2</math>, we denote by <math>\sin_p</math>. This is defined on the interval <math>[0,\pi_p/2]</math>, where <math>\tilde \pi_p</math> is defined as follows:

<math>\tilde \pi_p=2 \int_{0}^{1}(1-t^p)^{-(1/p)}\,dt</math>.

Because of this, we know that <math>\sin_p</math> is strictly increasing on <math>[0,\tilde \pi_p/2]</math>, <math>\sin_p(0)=0</math> and <math>\sin_p(\tilde \pi_p/2)=1</math>. We extend <math>\sin_p</math> to <math>[0,\tilde \pi_p]</math> by defining:

<math>\sin_p(x)=\sin_p(\tilde \pi_p-x)</math> for <math>x \in[\tilde \pi_p/2,\tilde \pi_p ]</math> Similarly <math>\cos_p(x)=(1-(\sin_p(x))^p)^\frac{1}{p}</math>.

Thus <math>\cos_p</math> is strictly decreasing on <math>[0,\tilde \pi_p/2]</math>, <math>\cos_p(0)=1</math> and <math>\cos_p(\tilde \pi_2/2)=0</math>. Also:

<math>|\sin_px|^p+|\cos_px|^p=1</math> .

This is immediate if <math>x \in [0,\tilde \pi/2 ]</math>, but it holds for all <math>x \in \R</math> in view of symmetry and periodicity.

== Applications == Squigonometric substitution can be used to solve indefinite integrals using a method akin to trigonometric substitution, such as integrals in the generic form<ref name="SHS">{{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 }}</ref> :<math>I = \int ({1-t^p})^\frac{1}{p}\,dt</math> 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}}

==See also== * Astroid * Ellipsoid * {{mvar|L^p}} spaces * Oval * Squround

== References == {{reflist}} Category:Trigonometry