Parabolic constant

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The ratio of the arc length of the parabolic segment formed by the latus rectum of any parabola to its focal parameter is a constant denoted \!\ P_2, which has the value:

\!\ P_2 = \sqrt2 + \ln(1 + \sqrt2) \approx 2.29558714939...
\!\ P_2 = \sqrt2 + \sinh^{-1}1
\!\ P_2 = \sqrt2 + \cosh^{-1}\sqrt2

Using the definitions of the hyperbolic functions, one also finds:

\!\ P_2 = \sqrt2 + \tanh^{-1}{\sqrt2 \over 2}
\!\ P_2 = \sqrt2 + {1 \over 2}\ln({{1+{\sqrt2 \over 2}} \over {1 - {\sqrt2 \over 2}}}) = \sqrt2 + {1 \over 2}\ln(2\sqrt2 + 3)

This leads back to the original expression because

\!\ \sqrt{2\sqrt2 + 3} = \sqrt2 + 1

The circle and parabola are unique among conic sections in that they have a universal constant, the analogous ratios for ellipses and hyperbolas depend on their eccentricities. This means that all circles and parabolas are similar, and that all ellipses and hyperbolas are not.

The distance from a point randomly selected in the unit square to its center is

d_{avg} = {{P_2} \over 6}

The Lindemann-Weierstrass theorem easily shows that \!\ P_2 is transcendental. A proof by contradiction:

Suppose that \!\ P_2 is algebraic. If this is true, then \!\ P_2 - \sqrt2 = \ln(1 + \sqrt2) must also be algebraic. However, by the Lindemann-Weierstrass theorem, \!\ e^{\ln(1+ \sqrt2)} = 1 + \sqrt2 would be transcendental, which is an obvious contradiction.

Since \!\ P_2 is transcendental, it is also irrational.

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