“Universal parabolic constant”的意思、由来-开放百科全书

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Universal parabolic constant

释义

Derivation
Properties
Applications
References and footnotes

The universal parabolic constant is a mathematical constant.

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. The focal parameter is twice the focal length. The ratio is denoted P.^[1]^[2]^[3]

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 of the parabola is the point F and the directrix is the line L.)

The value of P is^[4]

{{OEIS|id=A103710}}. 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 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.

Proof. Suppose that P is algebraic. Then must also be algebraic. However, by the Lindemann–Weierstrass theorem, would be transcendental, which is not the case. Hence P is transcendental.

Since P is transcendental, it is also irrational.

Applications

The average distance from a point randomly selected in the unit square to its center is^[5]

Proof.

References and footnotes

1. ^{{MathWorld||author=Sylvester Reese and Jonathan Sondow|title=Universal Parabolic Constant|urlname=UniversalParabolicConstant}}, a Wolfram Web resource.
2. ^{{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}}
3. ^{{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.
4. ^See Parabola#Length of an arc of a parabola. Use p=2f, the length of the semilatus rectum, so h=f and q=f.sqrt(2). Calculate 2s in terms of f, then divide by 2f, which is the focal parameter.
5. ^{{MathWorld||author=Eric W. Weisstein|title=Square Point Picking|urlname=SquarePointPicking}}, a Wolfram Web resource.

5 : Transcendental numbers|Mathematical constants|Conic sections|Parabolas|Real transcendental numbers

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