Universal Parabolic Constant

The universal parabolic constant is the ratio of the arc length of the parabolic segment formed by the latus rectum of any parabola to its semilatus rectum (and focal parameter). This ratio is independent of the parabola, just as the ratio of the arc length of a semicircle to its radius is always :

			(1)
			(2)
			(3)
			(4)

(OEIS A103710). For a parabolic segment of height and half-width , the full arc length is

(5)

Taking and gives , since the semilatus rectum is .

The other conic sections, namely the ellipse and hyperbola, do not have such universal constants because the analogous ratios for them depend on their eccentricities. In other words, all circles are similar and all parabolas are similar, but the same is not true for ellipses or hyperbolas (Ogilvy 1990, p. 84).

The area of the surface generated by revolving for about the -axis is given by

			(6)
			(7)

(Love 1950, p. 288; OEIS A103713) and the area of the surface generated by revolving for about the -axis is

			(8)
			(9)

(Love 1950, p. 288; OEIS A103714).

The expected distance from a uniformly selected point in the unit square to its center (square point picking) is

			(10)
			(11)

(Finch 2003, p. 479; OEIS A103712).

is an irrational number. It is also a transcendental number, as can be seen as follows. If were algebraic, then would also be algebraic. This logarithm is nonzero because . But then, by the Lindemann-Weierstrass theorem, would be transcendental, which is a contradiction.

The mean cylindrical radius about the -axis for a point chosen uniformly in the hemicube , cut from a unit cube, is .