# Squigonometric Sine Function

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The *-distance* from a point to the origin is defined as . In this geometry, the set of points satisfying can be considered a generalization of the Euclidean unit circle with , and with it comes a generalization of trigonometry ("squigonometry"). This Demonstration shows a graph of the -sine function alongside the -"squircle" with a red dot showing and the corresponding point on the squircle. The parameter is not the angle, but it is twice the area subtended by a squircular arc, shown when you click the "area" checkbox. Thus the area of the squircle is also half of the period of -sine, or a generalized from for . The value indicates the maximum distance , a limiting case dual to the taxicab metric . The cases and are also dual, while shows that the idea works even if the length does not satisfy the triangle inequality.

Contributed by: William E. Wood (January 2023)

Open content licensed under CC BY-NC-SA

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Reference

[1] R. D. Poodiack and William E. Wood, *Squigonometry: The Study of Imperfect Circles*, Cham: Springer, 2022.

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