Squigonometric Functions

The squigonometric functions are generalized sine and cosine functions that parameterize a unit superellipse using twice the area of the swept sector. For an integer , they are initially defined in the first quadrant by writing and for the unique solutions of the coupled initial-value problem

(1)

Differentiation then gives

(2)

so parameterizes the unit -circle . If denotes the area enclosed by this curve, then the first quadrant is traversed as runs from 0 to , and

			(3)
			(4)

where is the beta function. In particular, as .

The inverse squigonometric functions on are

			(5)
			(6)

Defining and gives

			(7)
			(8)

These identities make possible inverse squigonometric substitution. For example, Poodiack (2026) uses together with the beta function and reflection relation to obtain the improper integral

(9)

The squigonometric functions are distinct from the p-trigonometric functions. At , both systems trace the same unit superellipse, but the defining inverse integrand for the -sine has exponent instead of . Its half-period tends to 2, whereas the squigonometric constant is the area of the unit -circle and tends to 4. The two systems agree when .