Parabolic constant

In mathematics, 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 $!\; P\_2$.

In the diagram, the latus rectum is pictured in blue, the parabolic segment that it forms in red and the focal parameter in green.

The value of $!\; P\_2$ is $ln(1\; +\; sqrt2)\; +\; sqrt2\; approx\; 2.29558714939...$.

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.

Derivation

Take $y\; =\; frac\{x^2\}\{4a\}$ as the equation of the parabola. The focal parameter is $p=2a$ and the semilatus rectum is $l=2a$.

Properties

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.

Applications

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

:$d\_\{avg\}\; =$P_2} over 6}

External links

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