Squigonometry


Squigonometry or -trigonometry is a generalization of traditional trigonometry which replaces the circle and Euclidean distance function with the squircle and -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 in his 1990 pamphlet Creating Problems. 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.

Squigonometric functions

Cosquine and squine

Definition through unit squircle

The cosquine and squine functions, denoted as and 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. Here corresponds to and corresponds to
Notably, when, 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 by solving the coupled initial value problem
Where corresponds to and corresponds to.

Definition through analysis">mathematical analysis">analysis

The definition of sine and cosine through integrals can be extended to define the squigonometric functions. Let and define a differentiable function by:
Since is strictly increasing it is a one-to-one function on with range, where is defined as follows:
Let be the inverse of on. This function can be extended to by defining the following relationship:
By this means is differentiable in and, corresponding to this, the function is defined by:

Tanquent, cotanquent, sequent and cosequent

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

Inverse squigonometric functions

General versions of the inverse squine and cosquine can be derived from the initial value problem above. Let ; by the inverse function rule,. Solving for gives the definition of the inverse cosquine:
Similarly, the inverse squine is defined as:

Multiple ways to approach Squigonometry

Other parameterizations of squircles give rise to alternate definitions of these functions. For example, Edmunds, Lang, and Gurka define as:
Since is strictly increasing it has a =n inverse which, by analogy with the case, we denote by. This is defined on the interval, where is defined as follows:
.
Because of this, we know that is strictly increasing on, and. We extend to by defining:
for Similarly.
Thus is strictly decreasing on, and. Also:
.
This is immediate if, but it holds for all 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
that are otherwise computationally difficult to handle.
Squigonometry has been applied to find expressions for the volume of superellipsoids, such as the superegg.