Squigonometry
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.
:[1]: 99–100 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.
[1]: 100–101 The term squigonometry is a portmanteau of squircle and trigonometry.
The first use of the term "squigonometry" is undocumented: 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.
Nonetheless, it is well established that the idea of parametrizing curves that aren't perfect circles has been around for around 300 years.
[2] Over the span of three centuries, many mathematicians have thought about using functions similar to trigonometric functions to parameterize these generalized curves.
The cosquine and squine functions, denoted as
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: where
is a real number greater than or equal to 1.
, 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[3] by solving the coupled initial value problem[4][5] Where
[6] The definition of sine and cosine through integrals can be extended to define the squigonometric functions.
and define a differentiable function
is strictly increasing it is a one-to-one function on
This function can be extended to
by defining the following relationship: By this means
and, corresponding to this, the function
is defined by: The tanquent, cotanquent, sequent and cosequent functions can be defined as follows:[1]: 96 [7] General versions of the inverse squine and cosquine can be derived from the initial value problem above.
; by the inverse function rule,
{\displaystyle {\frac {dx}{dy}}=-[\operatorname {sq} _{p}(y)]^{p-1}=(1-x^{p})^{(p-1)/p}}
gives the definition of the inverse cosquine: Similarly, the inverse squine is defined as: Other parameterizations of squircles give rise to alternate definitions of these functions.
For example, Edmunds, Lang, and Gurka [8] define
is strictly increasing it has a =n inverse which, by analogy with the case
is strictly increasing on
in view of symmetry and periodicity.
Squigonometric substitution can be used to solve integrals, such as integrals in the generic form
