Squigonometry

Squigonometry or -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 -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.

Squigonometry or -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 -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. ===Cosquine and squine=== ====Definition through unit squircle==== The cosquine and squine functions, denoted as \operatorname{cq}_p(t) and \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: |x|^p + |y|^p = 1 where p is a real number greater than or equal to 1. Here x corresponds to \operatorname{cq}_p(t) and y corresponds to \operatorname{sq}_p(t) . Notably, when 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 by solving the coupled initial value problem \begin{cases} x'(t)=-|y(t)|^{p-1}\\ y'(t)=|x(t)|^{p-1}\\ x(0)=1\\ y(0)=0 \end{cases} Where x corresponds to \operatorname{cq}_p(t) and y corresponds to \operatorname{sq}_p(t) . ====Definition through analysis==== The definition of sine and cosine through integrals can be extended to define the squigonometric functions. Let 1 and define a differentiable function F_p:[0,1]\rightarrow by: F_p (x)=\int_{0}^{x}\frac{1}{{(1-t^p)}^\tfrac{p-1}{p}}\,dt Since F_p is strictly increasing it is a one-to-one function on [0,1] with range [0,\pi_p/2] , where \pi_p is defined as follows: \pi_p=2\int_{0}^{1}\frac{1}{{(1-t^p)}^\tfrac{p-1}{p}}\,dt Let \operatorname{sq}_p be the inverse of F_p on [0,\pi_p/2] . This function can be extended to [0,\pi_p] by defining the following relationship: \operatorname{sq}_p (x)=\operatorname{sq}_p (\pi_p-x) By this means \operatorname{sq}_p is differentiable in and, corresponding to this, the function \operatorname{cq}_p is defined by: \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: \operatorname{tq}_p(t)=\frac{\operatorname{sq}_p(t)}{\operatorname{cq}_p(t)} \operatorname{ctq}_p(t)=\frac{\operatorname{cq}_p(t)}{\operatorname{sq}_p(t)} \operatorname{seq}_p(t)=\frac{1}{\operatorname{cq}_p(t)} \operatorname{csq}_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 x=\operatorname{cq}_p (y) ; by the inverse function rule, \frac{dx}{dy} =-[\operatorname{sq}_p (y)]^{p-1}=(1-x^p)^{(p-1)/p} . Solving for y gives the definition of the inverse cosquine: 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: \operatorname{sq}_{p}^{-1}(x) = \int_{0}^{x}\frac{1}{(1-t^p)^{\frac{p-1}{p}}}\,dt \tilde F_p (x)= \int_{0}^{x}(1-t^p)^{-(1/p)}\,dt . Since F_p is strictly increasing it has an inverse which, by analogy with the case p=2 , we denote by \sin_p . This is defined on the interval [0,\pi_p/2] , where \tilde \pi_p is defined as follows: \tilde \pi_p=2 \int_{0}^{1}(1-t^p)^{-(1/p)}\,dt . Because of this, we know that \sin_p is strictly increasing on [0,\tilde \pi_p/2] , \sin_p(0)=0 and \sin_p(\tilde \pi_p/2)=1 . We extend \sin_p to [0,\tilde \pi_p] by defining: \sin_p(x)=\sin_p(\tilde \pi_p-x) for x \in[\tilde \pi_p/2,\tilde \pi_p ] Similarly \cos_p(x)=(1-(\sin_p(x))^p)^\frac{1}{p} . Thus \cos_p is strictly decreasing on [0,\tilde \pi_p/2] , \cos_p(0)=1 and \cos_p(\tilde \pi_2/2)=0 . Also: |\sin_px|^p+|\cos_px|^p=1 . This is immediate if x \in [0,\tilde \pi/2 ] , but it holds for all x \in \R in view of symmetry and periodicity. 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. ==See also== Astroid Ellipsoid spaces Oval Squround

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