Bow Curve

The bow curve is the quartic curve with implicit equation

(1)

It has parametric equations and , where is a real number. The intervals and trace the two loops. The bow curve has vertical tangents at and horizontal tangents at .

At points where , its signed curvature for the orientation with unit tangent vector is given by

(2)

where and and denote its partial derivatives. The origin is the only point of the curve where both partial derivatives vanish, so this implicit curvature formula applies when .

The area enclosed by the two loops is given by

			(3)
			(4)

(OEIS A118321).

The portion of the curve bounding the two loops has perimeter

			(5)
			(6)
			(7)

(OEIS A118322).

For an elliptic closed form, let be the real root and , the complex conjugate roots, with , of

(8)

Put , , , , and , using principal branches. In terms of Carlson elliptic integrals,

(9)

An equivalent Legendre form is

(10)

where , , and are elliptic integrals of the first, second, and third kinds, respectively, with elliptic modulus . The expression is real despite its complex parameters.