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| [[Image:Parabolic constant illustration v4.svg|right]]
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| The '''universal parabolic constant''' is a [[mathematical constant]]. | |
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| It is defined as the ratio, for any [[parabola]], of the [[arc length]] of the parabolic segment formed by the [[latus rectum]] to the [[focal parameter]]. It is denoted ''P''.<ref>{{MathWorld||author=Sylvester Reese and Jonathan Sondow|title=Universal Parabolic Constant|urlname=UniversalParabolicConstant}}, a Wolfram Web resource.</ref><ref>{{cite web|last=Reese|first=Sylvester|title=Pohle Colloquium Video Lecture: The universal parabolic constant|url=http://gaia.adelphi.edu/cgi-bin/makehtmlmov-css.pl?rtsp://gaia.adelphi.edu:554/General_Lectures/Pohle_Colloquiums/pohle200502.mov,pohle200502.mov,256,200|accessdate=February 2, 2005}}</ref><ref>{{cite arXiv |last=Sondow |first=Jonathan |eprint=1210.2279 |class=math.HO |title=The parbelos, a parabolic analog of the arbelos|year=2012}} [[American Mathematical Monthly]], 120 (2013), 929-935.</ref>
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| 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 (geometry)|focus]] of the parabola is the point ''F'' and the [[Conic_section#Eccentricity.2C_focus_and_directrix|directrix]] is the line ''L''.)
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| The value of ''P'' is
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| : <math>\ln(1 + \sqrt2) + \sqrt2 = 2.29558714939\dots</math>
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| {{OEIS|id=A103710}}. The [[circle]] and parabola are unique among conic sections in that they have a universal constant. The analogous ratios for [[ellipse]]s and [[hyperbola]]s depend on their [[Eccentricity (mathematics)|eccentricities]]. This means that all circles are [[Similarity (geometry)|similar]] and all parabolas are similar, whereas ellipses and hyperbolas are not.
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| ==Derivation==
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| Take <math>y = \frac{x^2}{4a}</math> as the equation of the parabola. The focal parameter is <math>p=2a</math> and the [[semilatus rectum]] is <math>\ell=2a</math>.
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| :<math>
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| \begin{align}
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| P & := \frac{1}{p}\int_{-\ell}^\ell \sqrt{1+\left(\frac{dy}{dx}\right)^2}\, dx \\
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| & = \frac{1}{2a}\int_{-2a}^{2a}\sqrt{1+\frac{x^2}{4a^2}}\, dx \\
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| & = \int_{-1}^{1}\sqrt{1+t^2}\, dt \quad (x=2at) \\
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| & = \operatorname{arcsinh}(1)+\sqrt{2}\\
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| & = \ln(1+\sqrt{2})+\sqrt{2}.
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| \end{align}
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| </math> | |
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| ==Properties==
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| ''P'' is a [[transcendental number]].
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| :'''Proof'''. Suppose that ''P'' is [[Algebraic number|algebraic]]. Then <math> \!\ P - \sqrt2 = \ln(1 + \sqrt2)</math> must also be algebraic. However, by the [[Lindemann–Weierstrass theorem]], <math> \!\ e^{\ln(1+ \sqrt2)} = 1 + \sqrt2 </math> would be transcendental, which is not the case. Hence ''P'' is transcendental. | |
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| Since ''P'' is transcendental, it is also [[irrational number|irrational]].
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| ==Applications==
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| The average distance from a point randomly selected in the unit square to its center is<ref>{{MathWorld||author=Eric W. Weisstein|title=Square Point Picking|urlname=SquarePointPicking}}, a Wolfram Web resource.</ref>
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| :<math> d_\text{avg} = {P \over 6}. </math>
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| ==References==
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| <references/>
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| [[Category:Transcendental numbers]]
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| [[Category:Mathematical constants]]
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| [[Category:Conic sections]]
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| [[Category:Parabolas]]
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