Squigonometry

Squigonometry or 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 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. ^[1] 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. ^[2]

Squigonometric functions

Cosquine and squine

Definition through unit squircle

Unit squircle for different values of p

The cosquine and squine functions, denoted as ${\displaystyle \operatorname {cq} _{p}(t)}$ and ${\displaystyle \operatorname {sq} _{p}(t),}$ 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:

${\displaystyle |x|^{p}+|y|^{p}=1}$

where ${\displaystyle p}$ is a real number greater than or equal to 1. Here ${\displaystyle x}$ corresponds to ${\displaystyle \operatorname {cq} _{p}(t)}$ and ${\displaystyle y}$ corresponds to ${\displaystyle \operatorname {sq} _{p}(t)}$

Notably, when ${\displaystyle p=2}$, 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 ^[3] by solving the coupled initial value problem ^{[4] [5]}

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

Where ${\displaystyle x}$ corresponds to ${\displaystyle \operatorname {cq} _{p}(t)}$ and ${\displaystyle y}$ corresponds to ${\displaystyle \operatorname {sq} _{p}(t)}$. ^[6]

Definition through analysis

The definition of sine and cosine through integrals can be extended to define the squigonometric functions. Let ${\displaystyle 1<p<\infty }$ and define a differentiable function ${\displaystyle F_{p}:[0,1]\rightarrow {\mathbb {R} }}$ by:

${\displaystyle F_{p}(x)=\int _{0}^{x}{\frac {1}{{(1-t^{p})}^{\tfrac {p-1}{p}}}}\,dt}$

Since ${\displaystyle F_{p}}$ is strictly increasing it is a one-to-one function on ${\displaystyle [0,1]}$ with range ${\displaystyle [0,\pi _{p}/2]}$, where ${\displaystyle \pi _{p}}$ is defined as follows:

${\displaystyle \pi _{p}=2\int _{0}^{1}{\frac {1}{{(1-t^{p})}^{\tfrac {p-1}{p}}}}\,dt}$

Let ${\displaystyle \operatorname {sq} _{p}}$ be the inverse of ${\displaystyle F_{p}}$ on ${\displaystyle [0,\pi _{p}/2]}$. This function can be extended to ${\displaystyle [0,\pi _{p}]}$ by defining the following relationship:

${\displaystyle \operatorname {sq} _{p}(x)=\operatorname {sq} _{p}(\pi _{p}-x)}$

By this means ${\displaystyle \operatorname {sq} _{p}}$ is differentiable in ${\displaystyle {\mathbb {R} }}$ and, corresponding to this, the function ${\displaystyle \operatorname {cq} _{p}}$ is defined by:

${\displaystyle {\frac {d}{dx}}\operatorname {sq} _{p}(x)=\operatorname {cq} _{p}(x)^{p-1}.}$

Tanquent, cotanquent, sequent and cosequent

The tanquent, cotanquent, sequent and cosequent functions can be defined as follows: ^{[7] [8]}

${\displaystyle \operatorname {tq} _{p}(t)={\frac {\operatorname {sq} _{p}(t)}{\operatorname {cq} _{p}(t)}}}$

${\displaystyle \operatorname {ctq} _{p}(t)={\frac {\operatorname {cq} _{p}(t)}{\operatorname {sq} _{p}(t)}}}$

${\displaystyle \operatorname {seq} _{p}(t)={\frac {1}{\operatorname {cq} _{p}(t)}}}$

${\displaystyle \operatorname {cseq} _{p}(t)={\frac {1}{\operatorname {sq} _{p}(t)}}}$

Inverse squigonometric functions

General versions of the inverse squine and cosquine can be derived from the initial value problem above. Let ${\displaystyle x=cq_{p}(y)}$; by the inverse function rule, ${\displaystyle {\frac {dx}{dy}}=-[\operatorname {sq} _{p}(y)]^{p-1}=(1-x^{p})^{(p-1)/p}}$. Solving for ${\displaystyle y}$ gives the definition of the inverse cosquine:

${\displaystyle y=\operatorname {cq} _{p}^{-1}(x)=\int _{x}^{1}{\frac {1}{(1-t^{p})^{\frac {p-1}{p}}}}\,dt}$

Similarly, the inverse squine is defined as:

${\displaystyle \operatorname {sq} _{p}^{-1}(x)=\int _{0}^{x}{\frac {1}{(1-t^{p})^{\frac {p-1}{p}}}}\,dt}$

Multiple ways to approach Squigonometry

Other parameterizations of squircles give rise to alternate definitions of these functions. For example, Edmunds, Lang, and Gurka ^[9] define ${\displaystyle {\tilde {F}}_{p}(x)}$ as:

${\displaystyle {\tilde {F}}_{p}(x)=\int _{0}^{x}(1-t^{p})^{-(1/p)}\,dt}$.

Since ${\displaystyle F_{p}}$ is strictly increasing it has a =n inverse which, by analogy with the case ${\displaystyle p=2}$, we denote by ${\displaystyle \sin _{p}}$. This is defined on the interval ${\displaystyle [0,\pi _{p}/2]}$, where ${\displaystyle {\tilde {\pi }}_{p}}$ is defined as follows:

${\displaystyle {\tilde {\pi }}_{p}=2\int _{0}^{1}(1-t^{p})^{-(1/p)}\,dt}$.

Because of this, we know that ${\displaystyle \sin _{p}}$ is strictly increasing on ${\displaystyle [0,{\tilde {\pi }}_{p}/2]}$, ${\displaystyle \sin _{p}(0)=0}$ and ${\displaystyle \sin _{p}({\tilde {\pi }}_{p}/2)=1}$. We extend ${\displaystyle \sin _{p}}$ to ${\displaystyle [0,{\tilde {\pi }}_{p}]}$ by defining:

${\displaystyle \sin _{p}(x)=\sin _{p}({\tilde {\pi }}_{p}-x)}$ for ${\displaystyle x\in [{\tilde {\pi }}_{p}/2,{\tilde {\pi }}_{p}]}$ Similarly ${\displaystyle \cos _{p}(x)=(1-(\sin _{p}(x))^{p})^{\frac {1}{p}}}$.

Thus ${\displaystyle \cos _{p}}$ is strictly decreasing on ${\displaystyle [0,{\tilde {\pi }}_{p}/2]}$, ${\displaystyle \cos _{p}(0)=1}$ and ${\displaystyle \cos _{p}({\tilde {\pi }}_{2}/2)=0}$. Also:

${\displaystyle |\sin _{p}x|^{p}+|\cos _{p}x|^{p}=1}$ .

This is immediate if ${\displaystyle x\in [0,{\tilde {\pi }}/2]}$, but it holds for all ${\displaystyle x\in \mathbb {R} }$ 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 ^[7]

${\displaystyle I=\int ({1-t^{p}})^{\frac {1}{p}}\,dt}$

that are otherwise computationally difficult to handle.

Squigonometry has been applied to find expressions for the volume of superellipsoids, such as the superegg. ^[7]

See also

• Astroid
• Ellipsoid
• L^p spaces
• Oval
• Squround

References

1. Holton, Derek (1990). Creating Problems: Counting; Packing; Intersecting; Chessboards; Squigonometry. Derek Holton's problem solving series. Vol. 15. University of Otago. ISBN   0-908903-15-4. Reprised in Holton, Derek (2011). "Squigonometry". A Second Step to Mathematical Olympiad Problems. Singapore: World Scientific. § 7.6, pp. 233–235. doi:10.1142/7979. ISBN   978-981-4327-87-9.
2. Poodiack, Robert D.; Wood, William E. (2022). Squigonometry: The Study of Imperfect Circles. Springer. p. 1. doi:10.1007/978-3-031-13783-9. ISBN   978-3-031-13782-2.
   Examples:

   Lundberg, E. (1879). Om hypergoniometriska funktioner af komplexa variabla (Manuscript). Translation by Jaak Peetre (2000) "On hypergoniometric functions of complex variables" (Postscript file).

   Shelupsky, D. (1959). "A generalization of the trigonometric functions". The American Mathematical Monthly. 66 (10): 879–884. JSTOR   2309789.
3. Elbert, Á. (1987-09-01). "On the half-linear second order differential equations" . Acta Mathematica Hungarica. 49 (3): 487–508. doi:10.1007/BF01951012. ISSN   1588-2632.
4. Wood, William E. (October 2011). "Squigonometry". Mathematics Magazine. 84 (4): 264.
5. Chebolu, Sunil; Hatfield, Andrew; Klette, Riley; Moore, Cristopher; Warden, Elizabeth (Fall 2022). "Trigonometric functions in the p-norm". BSU Undergraduate Mathematics Exchange. 16 (1): 4, 5.
6. Girg, Petr E.; Kotrla, Lukáš (February 2014). Differentiability properties of p-trigonometric functions. p. 104.
7. Poodiack, Robert D. (April 2016). "Squigonometry, Hyperellipses, and Supereggs" . Mathematics Magazine. 89 (2): 92–102. doi:10.4169/math.mag.89.2.92.
8. Edmunds, David E.; Gurka, Petr; Lang, Jan (2012). "Properties of generalized trigonometric functions". Journal of Approximation Theory. 164 (1): 49.
9. Edmunds, David (2011). Eigenvalues, Embeddings and Generalised Trigonometric Functions. Springer-Verlag Berlin Heidelberg.