[math-fun] area inside algebraic curve impossible to express?
For an explicit impossibility conjecture, consider the finite-area loop containing (-2,1) bounded by the fairly-generic-looking cubic curve 3x^3 + 4x^2y + 7y^2x - 10y^3 - 36x + 37y = 41. I doubt this area, which is approximately 4.68450311946587, can be expressed in closed form. [Number found using polar coordinates centered at (-2.2, 1.1); integrate (r^2/2) dtheta using trapezoidal rule. All digits probably correct.] If anybody recomputed this to, say, 50 decimals, then huge runs of number-guesser tools could partially confirm the conjecture, I guess. Ries -l6 realizes it has nothing & gives up in disgust. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
I think the question is interesting. I had the same idea -- compute some sample areas and use RIES -- but nothing turned up. Charles Greathouse Analyst/Programmer Case Western Reserve University On Thu, Dec 11, 2014 at 7:36 PM, Warren D Smith <warren.wds@gmail.com> wrote:
For an explicit impossibility conjecture, consider the finite-area loop containing (-2,1) bounded by the fairly-generic-looking cubic curve
3x^3 + 4x^2y + 7y^2x - 10y^3 - 36x + 37y = 41.
I doubt this area, which is approximately 4.68450311946587, can be expressed in closed form. [Number found using polar coordinates centered at (-2.2, 1.1); integrate (r^2/2) dtheta using trapezoidal rule. All digits probably correct.]
If anybody recomputed this to, say, 50 decimals, then huge runs of number-guesser tools could partially confirm the conjecture, I guess. Ries -l6 realizes it has nothing & gives up in disgust.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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On Thu, Dec 11, 2014 at 6:36 PM, Warren D Smith <warren.wds@gmail.com> wrote:
For an explicit impossibility conjecture, consider the finite-area loop containing (-2,1) bounded by the fairly-generic-looking cubic curve
3x^3 + 4x^2y + 7y^2x - 10y^3 - 36x + 37y = 41.
I doubt this area, which is approximately 4.68450311946587,
[snip] For this area I get using maple and no polar coordinates: 4.68450311946587007550715114328001623903733966001661250115501563297 if that is of any use to anyone. Probably all digits are correct.
I would expect that the area of the region is given by an elliptic integral. Use Green's theorem to show that the area is integral y dx where the integral is over the path of the bounding curve. The bounding curve is binational to an elliptic curve, and so is parametrized by elliptic functions. Victor On Fri, Dec 12, 2014 at 12:41 AM, James Buddenhagen <jbuddenh@gmail.com> wrote:
On Thu, Dec 11, 2014 at 6:36 PM, Warren D Smith <warren.wds@gmail.com> wrote:
For an explicit impossibility conjecture, consider the finite-area loop containing (-2,1) bounded by the fairly-generic-looking cubic curve
3x^3 + 4x^2y + 7y^2x - 10y^3 - 36x + 37y = 41.
I doubt this area, which is approximately 4.68450311946587,
[snip]
For this area I get using maple and no polar coordinates: 4.68450311946587007550715114328001623903733966001661250115501563297 if that is of any use to anyone. Probably all digits are correct. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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Charles Greathouse -
James Buddenhagen -
Victor Miller -
Warren D Smith