Area and Arclength of x^p+y^p=1
Shelley Walsh
Virgil
<14953316.1180718998...@nitrogen.mathforum.org>,
Shelley Walsh <shelle...@mac.com> wrote:
For odd p, it should be |x|^p + |y|^p = 1, at least if you want symmetry
about both axes.
Then p = 1 and p = oo are easy enough, a diamond and a square.
quasi
wrote:
>When p=2 you all know this is the circle, which we all know everything in the world about. I stumbled across a reason to want to know if there are any formula for these when p is not 2. The integrals are quite messy. I can do series approximation, but the seem to be simple enough shapes that it would seem that something should be already known. But a quick search didn't reveal anything obvious, so I thought I'd see if any of you knew anything.
I doubt the integrals can be solved in closed form for any p>2.
quasi
Shelley Walsh
David W. Cantrell
The area bounded by the curve in the first quadrant is
Gamma(1 + 1/p) Gamma(1/p)/(2 Gamma(2/p))
but I suspect that the arc length cannot be given in closed form in terms
of familiar functions.
David
Shelley Walsh
But what about the definite integrals? Is there any way to bypass the lack of elementary antiderivative to get formulas for them?
Dave L. Renfro
There are several old American Mathematical Monthly
papers that deal with these and other issues for
the curve x^p + y^p = 1 (center of mass, moments
of intertia, etc.). I have copies of the papers
at home and don't have JSTOR access to look up the
precise references now, but if you or anyone is
interested, I can post the details this weekend.
I think the papers appeared before 1920. If you
have a college library nearby, you could probably
find them in 20 minutes if the early volumes are
on the shelves (and not off somewhere in storage).
I could also e-mail them to you, but it may take me
a few days before I visit a library where this would
be possible.
Dave L. Renfro
Randy Hudson
Shelley Walsh <shelle...@mac.com> wrote:
I believe Piet Hein worked with these "superellipses"; I think there was an
early-70s Martin Gardner "Mathematical Recreations" column discussing his
work.
--
Randy Hudson
A N Niel
> but I suspect that the arc length cannot be given in closed form in terms
> of familiar functions.
If that arc length were known in simple terms, then it would
be in all the calculus textbooks... the exercises in the section on
arc length in existing texts are *very few* in number...
quasi
<DWCan...@sigmaxi.net> wrote:
>quasi <qu...@null.set> wrote:
>> On Fri, 01 Jun 2007 13:29:28 EDT, Shelley Walsh <shelle...@mac.com>
>> wrote:
>>
>> >When p=2 you all know this is the circle, which we all know everything
>> >in the world about. I stumbled across a reason to want to know if there
>> >are any formula for these when p is not 2. The integrals are quite
>> >messy. I can do series approximation, but the seem to be simple enough
>> >shapes that it would seem that something should be already known. But a
>> >quick search didn't reveal anything obvious, so I thought I'd see if any
>> >of you knew anything.
>>
>> I doubt the integrals can be solved in closed form for any p>2.
>
>The area bounded by the curve in the first quadrant is
>
>Gamma(1 + 1/p) Gamma(1/p)/(2 Gamma(2/p))
Cool.
>but I suspect that the arc length cannot be given in closed form in terms
>of familiar functions.
It's a surprise that the area to came out so nice.
quasi
David W. Cantrell
I agree.
BTW, when I said the bit quoted above, I was meaning that the arc length
cannot _generally_ be given in closed form in terms of familiar functions.
Of course, I did not mean to preclude its being given in closed form in
terms of familiar functions for _specific values of p_. Besides the
obviously simple cases in which this can be done (such as p = 1 or 2), here
are two others:
The arc length of the curve in the first quadrant is
3/2 when p = 2/3
and
1 + log(1 + sqrt(2))/sqrt(2) when p = 1/2.
David
galathaea
> On 01 Jun 2007 18:49:08 GMT, David W. Cantrell
> >quasi <q...@null.set> wrote:
> >> On Fri, 01 Jun 2007 13:29:28 EDT, Shelley Walsh <shelleywa...@mac.com>
> >> wrote:
>
> >> >When p=2 you all know this is the circle, which we all know everything
> >> >in the world about. I stumbled across a reason to want to know if there
> >> >are any formula for these when p is not 2. The integrals are quite
> >> >messy. I can do series approximation, but the seem to be simple enough
> >> >shapes that it would seem that something should be already known. But a
> >> >quick search didn't reveal anything obvious, so I thought I'd see if any
> >> >of you knew anything.
>
> >> I doubt the integrals can be solved in closed form for any p>2.
>
> >The area bounded by the curve in the first quadrant is
>
> >Gamma(1 + 1/p) Gamma(1/p)/(2 Gamma(2/p))
>
> Cool.
>
> >but I suspect that the arc length cannot be given in closed form in terms
> >of familiar functions.
>
> It's a surprise that the area to came out so nice.
its just a beta integral with a simple factoring
cf.
http://mathworld.wolfram.com/BetaIntegral.html
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
Badger
wrote:
>When p=2 you all know this is the circle, which we all know everything in the world about. I stumbled across a reason to want to know if there are any formula for these when p is not 2. The integrals are quite messy. I can do series approximation, but the seem to be simple enough shapes that it would seem that something should be already known. But a quick search didn't reveal anything obvious, so I thought I'd see if any of you knew anything.
Perhaps of interest:
<http://mathworld.wolfram.com/Superellipse.html>
<http://en.wikipedia.org/wiki/Superellipse>
quasi
wrote:
>On Jun 1, 1:33 pm, quasi <q...@null.set> wrote:
>> On 01 Jun 2007 18:49:08 GMT, David W. Cantrell
>> <DWCantr...@sigmaxi.net> wrote:
>> >quasi <q...@null.set> wrote:
>> >> On Fri, 01 Jun 2007 13:29:28 EDT, Shelley Walsh <shelleywa...@mac.com>
>> >> wrote:
>>
>> >> >When p=2 you all know this is the circle, which we all know everything
>> >> >in the world about. I stumbled across a reason to want to know if there
>> >> >are any formula for these when p is not 2. The integrals are quite
>> >> >messy. I can do series approximation, but the seem to be simple enough
>> >> >shapes that it would seem that something should be already known. But a
>> >> >quick search didn't reveal anything obvious, so I thought I'd see if any
>> >> >of you knew anything.
>>
>> >> I doubt the integrals can be solved in closed form for any p>2.
>>
>> >The area bounded by the curve in the first quadrant is
>>
>> >Gamma(1 + 1/p) Gamma(1/p)/(2 Gamma(2/p))
>>
>> Cool.
>>
>> >but I suspect that the arc length cannot be given in closed form in terms
>> >of familiar functions.
>>
>> It's a surprise that the area came out so nice.
>
>its just a beta integral with a simple factoring
>
>cf.
>
>http://mathworld.wolfram.com/BetaIntegral.html
Ah, ok -- thanks.
But I'd be really, really surprised if anyone can find a closed form
expression for the volume in the first orthant of the super-ellipsoid
x^p + y^p + z^p = 1
for any p>1 with the exception of p=2 (and maybe p=3/2).
quasi
quasi
Might as well make that:
for any p>0 with the exceptions p=1, p=2 (and the possible exceptions
p=1/2, p=3/2).
quasi
tommy1729
well they can
and they cannot
not in conventional form i believe.
but in many sorts of "super-duper exotic special functions"
hypertrinomials and so ...
i agree with you.
however a strong proof would ammaze me too.
ive posted a similar question many years ago , and didnt get an answer ( i mean no solution ).
regards
tommy1729
David W. Cantrell
Not so. And the result is quite beautiful and surely well known already:
Let x! be defined as Gamma(x + 1) for positive x.
Then the area in the first quadrant which I had given earlier in this
thread can be expressed, now more nicely, as
((1/p)!)^2 / (2/p)!
The volume in the first octant which you asked about above can be expressed
as
((1/p)!)^3 / (3/p)!
The generalization is that the corresponding n-volume is
((1/p)!)^n / (n/p)!
David W. Cantrell
Badger
[snip]
>But I'd be really, really surprised if anyone can find a closed form
>expression for the volume in the first orthant of the super-ellipsoid
>
> x^p + y^p + z^p = 1
>
>for any p>1 with the exception of p=2 (and maybe p=3/2).
>
>quasi
tommy1729
nice :-)
quasi
<DWCan...@sigmaxi.net> wrote:
Nice.
I guess I should have tried a simple test case such as n=3, p=3.
Let me see if Maple can do it ...
input to Maple:
p:=3;
Int(Int(Int(1,z=0..(1-x^p-y^p)^(1/p)),y=0..(1-x^p)^(1/p)),x=0..1);
v:=value(%);
output from Maple:
v:=8/243*Pi^3/GAMMA(2/3)^3*3^(1/2)
Yup, closed form, Maple has no problem doing it.
Let me check Maple's answer against the formula you specified ...
input to Maple:
w:=((GAMMA(1+1/p))^3)/GAMMA(1+(3/p));
simplify(v-w);
output from Maple:
0
Yup, everything checks.
I have to admit, I'm too lazy to do this stuff by hand.
Well, the closed form was definitely a surprise to me.
Thanks.
quasi
quasi
wrote:
Thanks for the link.
So much for my guess.
It seems that every p>0 wants to be an exception.
quasi
Larry Hammick
> I believe Piet Hein worked with these "superellipses"; I think there was
> an
> early-70s Martin Gardner "Mathematical Recreations" column discussing his
> work.
x^(5/2) + y^(5/2) = constant
is used in some of Piet Hein's architecture. Go here:
http://www.piethein.com/usr/piethein/HomepagUK.nsf
and click on "superellipse". You get Gardner's article and a photo of a
traffic "circle" of that shape.
LH
Dave L. Renfro
> There are several old American Mathematical Monthly
> papers that deal with these and other issues for
> the curve x^p + y^p = 1 (center of mass, moments
> of intertia, etc.). I have copies of the papers
> at home and don't have JSTOR access to look up the
> precise references now, but if you or anyone is
> interested, I can post the details this weekend.
Below are the details and references. I've also included
a number of references about super-ellipses -- references
that I think it would be useful to archive in this thread.
-------------------------------------------------------------
Michael Brozinsky and Murray S. Klamkin, "Solution to
Problem #4495", School Science and Mathematics 95 #8
(December 1995), 442.
Problem proposed by Richard L. Francis: "Although it is
impossible to square the circle x^2 + y^2 = 1, is it
possible to square [the] hypocycloid x^(2/3) + y^(2/3) =1?"
Solution I: Brozinsky shows [1] that the area enclosed by
the curve is (pi)(n-1)(n-2) / n^2, so the answer is "no".
[n = 4, which has the name "astroid", gives the curve asked for]
[1] Start with parametric equations for a hypocycloid of
n cusps formed by a point on a circle of radius 1/n
rolling inside the unit circle, then use Green's theorem.
Solution II: Klamkin shows that the area within x^p + y^p = a^p
is ("by Dirichlet's integral"):
{ (a^2) * [Gamma(1/p)]^2 } / { p^2 * Gamma(1 + 2/p) }
Klamkin goes on to say that if p = 2/3, then we get 3*pi*a^2 / 32,
and so the answer is "no". [I believe there is a typo somewhere,
because Brozinsky's result (a = 1) gives (3/8) * pi.
-------------------------------------------------------------
C. C. Yenn, "Solution to Calculus Problem #419", American
Mathematical Monthly 24 #7 (September 1917), 332-333.
Proposed by C. C. Yenn: "Find the entire area of the surface
x^(2/3) + y^(2/3) + z^(2/3) = a^(2/3)."
The area is found to be (17/12) * pi * a^2.
-------------------------------------------------------------
George B. McClellan Zerr, "Note on areas and volumes",
American Mathematical Monthly 1 #11 (November 1894), 380-381.
Note: Zerr died unexpectedly on 7 October 1910. A short
biography (by Benjamin Franklin Finkel) appears in American
Mathematical Monthly 18 #1 (January 1911), 1-2.
Let m,n,p and a,b,c be positive real numbers. [Zerr initially says
that m,n,p are to be positive integers, but at the end of the paper
he says the formulas "will do for any admissible values of m,n,p".]
*************
The area within the curve
(x/a) ^ [2 / (m+1)] + (y/b) ^ [2 / (n+1)] = 1
is ("by Dirichlet's theorem")
{ ab(2m+1)(2n+1) * Gamma(m + 1/2) * Gamma(n + 1/2) } / Gamma(m+n+2).
SPECIAL CASES for a = b = 1:
m = 0 and n = 0 gives x^2 + y^2 = 1
area = 1
m = 1 and n = 1 gives x^(2/3) + y^(2/3) = 1
area = (3/8) * pi
m = 0 and n = 1 gives x^2 + y^(2/3) = 1
area = (3/4) * pi
m = 2 and n = 2 gives x^(2/5) + y^(2/5) = 1
area = (15/128) * pi
m = 1 and n = 2 gives x^(2/3) + y^(2/5) = 1
area = (15/64) * pi
m = 4 and n = 4 gives x^(2/9) + y^(2/9) = 1
area = (315 / 2^15) * pi
m = 3/2 and n = 3/2 gives x^(1/2) + y^(1/2) = 1
area = 2/3
*************
The volume within the surface
(x/a)^[2 / (m+1)] + (y/b)^[2 / (n+1)] + (z/c)^[2 / (p+1)] = 1
is:
{ abc(2m+1)(2n+1)(2p+1) * Gamma(m + 1/2) * Gamma(n + 1/2) * Gamma(p +
1/2) }
divided by Gamma(m + n + p + 5/2)
-------------------------------------------------------------
George B. McClellan Zerr, "The centroid of areas and volumes",
American Mathematical Monthly 3 #2 (February 1896), 46-49.
The center of mass within the curve above is found for
density functions of the form rho(x,y) = (x^r)*(y^s),
where r,s are positive real numbers.
-------------------------------------------------------------
George B. McClellan Zerr, "The centroid of areas and volumes",
American Mathematical Monthly 3 #3 & #4 (March & April 1896),
73-76 & 100-103. [Errata: AMM 3 #5 (May 1896), p. 158.]
The center of mass within the surface above is found for
density functions of the form rho(x,y,z) = (x^r)*(y^s)*(z^t),
where r,s,t are positive real numbers.
-------------------------------------------------------------
George B. McClellan Zerr, "Moments of inertia", American
Mathematical Monthly 4 #12 (December 1897), 303-306.
For a constant planar density, the moments of inertia about
the x-axis and about the y-axis are found for the region
within the curve above.
For a constant volume density, sufficient calculation is
given so that the 9 moments I_xx, I_xy, I_xz, ..., I_zz
can be easily written down.
At the end, Zerr points out that the radius of gyration
(for the region and for the volume, each with constant
density) can now be easily written down.
-------------------------------------------------------------
Regarding "super-ellipses", below are some references I
assembled about 18 months ago when I working on something
marginally related to this.
J. S. Forsyth and Zdzislaw Alexander Melzak, "Polyconics 1.
Polyellipses and optimization", Quarterly of Applied
Mathematics 35 (1977), 239-255. [MR 56 #7188; Zbl 367.50010]
Martin Gardner, "Fun with eggs: uncooked, cooked and mathematical",
Mathematical Games column, Scientific American 242 #4 (April 1980),
19, 20, 22, 24, 27, 178.
H. Gwynedd Green, "On some general ovals of Cassinian type",
Mathematical Gazette 27 #273 (February 1943), 4-12.
Ilya Kuznetsov, "The curve consisting of parts of ellipses",
Reader Reflections column, Mathematics Teacher 88 #7
(October 1995), 540 & 542. [Follow-up by Zerger below.]
James Clerk Maxwell, "On the description of oval curves, and
those having a plurality of foci; with remarks by Professor
Forbes", Proceedings of the Royal Society of Edinburgh 2 #28
(1845-46), 89-91.
Dated April 1846. Reprinted on 1-3 of William Davidson
Niven (editor), "The Scientific Papers of James Clerk
Maxwell", Volume I, Cambridge University Press, 1890.
Note: Forbes actually read the paper before the Royal
Society of Edinburgh (on 6 April 1846) because Maxwell
was not allowed due to his age. Maxwell was only
14 years old at the time.
Paul L. Rosin, "On Serlio's construction of ovals",
Mathematical Intelligencer 23 #1 (2001), 58-69.
http://users.cs.cf.ac.uk/Paul.Rosin/resources/papers/oval2.pdf
Paul L. Rosin, "On the construction of ovals", Proceedings of
The International Society of the Arts, Mathematics, and
Architecture (ISAMA), 2004, 118-122.
http://users.cs.cf.ac.uk/Paul.Rosin/resources/papers/ovalISAMA.pdf
Paul L. Rosin -- See "Ellipses, Superellipses, Ovals, etc." at
http://users.cs.cf.ac.uk/Paul.Rosin/papers.html
P. V. Sahadevan, "The theory of egglipse--a new curve with three
focal points", International Journal of Mathematical Education
in Science and Technology 18 (1987), 29-39.
[MR 88b:51041; Zbl 613.51030]
Junpei Sekino, "n-ellipses and the minimum distance sum problem",
American Mathematical Monthly 106 #3 (March 1999), 193-202.
[MR 2000a:52003; Zbl 986.51040]
Adrienne Stiff, "The generalized ellipse", student research
project (under Derrick J. Hylton?), physics department,
Spelman College, 1996.
Milton H. Sussman, "Maxwell's ovals and the refraction of
light", American Journal of Physics 34 (1966), 416-418.
Monte J. Zerger, "Maxwell's Easter eggs", Reader Reflections
column, Mathematics Teacher 90 #4 (April 1997), 260 & 312.
[Follow-up to Kuznetsov above.]
-------------------------------------------------------------
Dave L. Renfro
Gerry Myerson
<14953316.1180718998...@nitrogen.mathforum.org>,
Shelley Walsh <shelle...@mac.com> wrote:
> When p=2 you all know this is the circle, which we all know everything in the
> world about.
Hardly. See, e.g.,
http://mathworld.wolfram.com/GausssCircleProblem.html
http://mathworld.wolfram.com/CirclePacking.html
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
tommy1729
> <14953316.1180718998...@nitrogen.math
> forum.org>,
> Shelley Walsh <shelle...@mac.com> wrote:
>
> > When p=2 you all know this is the circle, which we
> all know everything in the
> > world about.
>
> Hardly. See, e.g.,
> http://mathworld.wolfram.com/GausssCircleProblem.html
> http://mathworld.wolfram.com/CirclePacking.html
>
> --
> Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for
> email)
i dont agree.
i see a lot of closed form solutions
expressesions in terms of number theory functions
etc
pretty complete id say.
tommy1729