Squigonometry or -trigonometry is a branch of mathematics that extends traditional trigonometry to shapes other than circles, particularly to squircles, in the -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] 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]
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 other curves that lack the circle’s perfection 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
cqp(t)
sqp(t),
|x|p+|y|p=1
p
x
cqp(t)
y
sqp(t)
Notably, when
p=2
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]
\begin{cases} x'(t)=-[y(t)]p-1\\ y'(t)=[x(t)]p-1\\ x(0)=1\\ y(0)=0 \end{cases}
x
cqp(t)
y
sqp(t)
The definition of sine and cosine through integrals can be extended to define the squigonometric functions. Let
1<p<infty
Fp:[0,1] → {{\R}}
Fp
| x | |
| (x)=\int | |
| 0 |
| 1 | |
| \sqrt[p]{1-tp |
Fp
[0,1]
[0,\pip/2]
\pip
\pip=2\int
| 1 | |
| 0 |
| 1 | |
| \sqrt[p]{1-tp |
sqp
Fp
[0,\pip/2]
[0,\pip]
sqp(x)=sqp(\pip-x)
sqp
{{\R}}
cqp
cqp(x)=
| d | |
| dx |
sqp(x)
The tanquent, cotanquent, sequent and cosequent functions can be defined as follows[1] :[7]
| tq | ||||
|
| ctq | ||||
|
| seq | ||||
|
| cseq | ||||
|
General versions of the inverse squine and cosquine can be derived from the initial value problem above. Let
x=cqp(y)
| dx | |
| dy |
=-[sqp(y)]p-1=(1-xp)(p-1)/p
y
| -1 | |
| y=cq | |
| p |
(x)=
| 1 | ||
| \int | ( | |
| x |
| 1 | |
| 1-tp |
| ||||
| ) |
dt
| -1 | |
| sq | |
| p |
(x)=
| x | ||
| \int | ( | |
| 0 |
| 1 | |
| 1-tp |
| ||||
| ) |
dt
Squigonometric substitution can be used to solve integrals, such as integrals in the generic form
I=
| b | |
| \int | |
| a |
({1-tp})
| ||||
dt