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Squigonometry

From Wikipedia, the free encyclopedia

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 

Etymology

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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.

Squigonometric functions

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Cosquine and squine

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Definition through unit squircle

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Unit squircle for different values of p

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

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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 corresponds to and corresponds to .[6]

Definition through analysis

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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

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The tanquent, cotanquent, sequent and cosequent functions can be defined as follows[1]: 96 :[7]

Inverse squigonometric functions

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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:

Applications

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Solving integrals

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Squigonometric substitution can be used to solve integrals, such as integrals in the generic form .

See also

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References

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  1. ^ a b c Poodiack, Robert D. (April 2016). "Squigonometry, Hyperellipses, and Supereggs". Mathematics Magazine. 89 (2).
  2. ^ Poodiack, Robert D.; Wood, William E. (2022). Squigonometry: The Study of Imperfect Circles (1st ed.). Springer Nature Switzerland. p. 1.
  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. ^ Edmunds, David E.; Gurka, Petr; Lang, Jan (2012). "Properties of generalized trigonometric functions". Journal of Approximation Theory. 164 (1): 49.