The universal parabolic constant is a
mathematical constant.
It is defined as the ratio, for any
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
, of the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
of the parabolic segment formed by the
latus rectum
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
to the focal parameter. The focal parameter is twice the
focal length. The ratio is denoted ''P''.
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
focus
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
of the parabola is the point ''F'' and the
directrix is the line ''L''.)
The value of ''P'' is
:
. The
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
and parabola are unique among conic sections in that they have a universal constant. The analogous ratios for
ellipses and
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s depend on their
eccentricities. This means that all circles are
similar and all parabolas are similar, whereas ellipses and hyperbolas are not.
Derivation
Take
as the equation of the parabola. The focal parameter is
and the
semilatus rectum is
.
Properties
''P'' is a
transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
.
:Proof. Suppose that ''P'' is
algebraic. Then
must also be algebraic. However, by the
Lindemann–Weierstrass theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:
In other words, the extension field \mathbb(e^, \dots, e^) has transce ...
,
would be transcendental, which is not the case. Hence ''P'' is transcendental.
Since ''P'' is transcendental, it is also
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
.
Applications
The average distance from a point randomly selected in the unit square to its center is
[, a Wolfram Web resource.]
:
:Proof.
:
References and footnotes
Mathematical constants
Conic sections
Parabolas
Real transcendental numbers