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Squigonometry

Squigonometry is the study of analogues of trigonometry associated with superellipses in place of circles (Wood 2011, Poodiack 2016, Poodiack and Wood 2022). For p>1, the corresponding unit superellipse is

|x|^p+|y|^p=1.

The p-trigonometric functions and squigonometric functions both give natural parametric equations (x,y)=(cos_pt,sin_pt), but they use different parameters. The former are defined by an inverse integrand with exponent -1/p, whereas the latter use twice the swept sector area and an inverse integrand with exponent -(p-1)/p. They coincide when p=2 but otherwise give different functions and different constants denoted pi_p (Poodiack 2026). These functions are central tools of squigonometry, but the subject also includes the geometry and applications of squircles and supereggs.

See also

Circle, p-Trigonometric Functions, Squigonometric Functions, Squircle, Superegg, Superellipse, Trigonometry

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References

Poodiack, R. D. "Squigonometry, Hyperellipses, and Supereggs." Math. Mag. 89, 92-102, 2016. https://doi.org/10.4169/math.mag.89.2.92.Poodiack, R. D. "A Squigonometric Way to Skin a Sequence of Definite Integrals." College Math. J., 1-12, 2026. https://doi.org/10.1080/07468342.2026.2702260.Poodiack, R. D. and Wood, W. E. Squigonometry: The Study of Imperfect Circles. Cham, Switzerland: Springer, 2022. https://doi.org/10.1007/978-3-031-13783-9.Wood, W. E. "Squigonometry." Math. Mag. 84, 257-265, 2011. https://doi.org/10.4169/math.mag.84.4.257.

Cite this as:

Weisstein, Eric W. "Squigonometry." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Squigonometry.html

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