Draft:Squigonometry

• Comment: A single (working) source isn't enough to establish notability, or even properly to verify the information. DoubleGrazing (talk) 11:35, 13 July 2024 (UTC)

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Squigonometry or p-trigonometry is a branch of mathematics that extends traditional trigonometry to shapes other than circles, particularly to squircles, in the p-norm. Unlike trigonometry, which deals with the relationships between angles and side lengths of triangles and uses trigonometric functions, squigonometry focuses on analogous relationships within the context of a unit squircle.

Squigonometric functions are mostly used to solve certain indefinite integrals, using a method akin to trigonometric substitution. This approach allows for the integration of functions that are otherwise computationally difficult to handle.

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

Etymology

The term squigonometry is a portmanteau of squircle and trigonometry. The term "squigonometry" was likely first used in the first decade of the 21st century: the coining of the word possibly emerged from mathematical curiosity and the need to solve problems involving squircle geometries. As mathematicians sought to generalize trigonometric ideas beyond circular shapes, they naturally extended these concepts to squircles, leading to the creation of new functions.

Squigonometric functions

Cosquine and squine

Definition through unit squircle

Unit squircle for different values of p

The cosquine and squine functions, denoted as $cq_{p}(t)$ and $sq_{p}(t),$ 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 $cq_{p}(t)$ and $y$ corresponds to $sq_{p}(t)$

Notably, when $p=2$, the squigonometric functions coincide with the trigonometric functions.

Definition through CIVP

Similarly to how trigonometric functions are defined through differential equations, the cosquine and squine functions are also uniquely determined by solving the coupled initial value problem (derived from differentiating the unit squircle):

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

Where $x$ corresponds to $cq_{p}(t)$ and $y$ corresponds to $sq_{p}(t)$.

Tanquent, cotanquent, sequent and cosequent

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

$tq_{p}(t)={\frac {sq_{p}(t)}{cq_{p}(t)}}$

$ctq_{p}(t)={\frac {cq_{p}(t)}{sq_{p}(t)}}$

$seq_{p}(t)={\frac {1}{cq_{p}(t)}}$

$cseq_{p}(t)={\frac {1}{sq_{p}(t)}}$

Inverse squigonometric functions

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, ${\frac {dx}{dy}}=-[sq_{p}(y)]^{p-1}=(1-x^{p})^{(p-1)/p}$. By solving for $y$ we get:

$y=cq_{p}^{-1}=\int _{x}^{1}({\frac {1}{1-t^{p}}})^{(p-1)/p}\,dt$

Similarly we have:

$sq_{p}^{-1}=\int _{0}^{x}({\frac {1}{1-t^{p}}})^{(p-1)/p}\,dt$

Applications

Solving integrals

Squigonometric substitution can be used to solve integrals, such as integrals in the generic form $I=\int _{a}^{b}({1-t^{p}})^{1/p}\,dt$.