cp's OEIS Frontend

A197008 Decimal expansion of the shortest distance from x axis through (1,2) to y axis.

%I A197008 #31 Feb 03 2025 09:35:49
%S A197008 4,1,6,1,9,3,8,1,8,4,9,4,1,4,6,2,7,5,2,3,9,0,0,8,0,7,2,2,9,4,6,6,9,9,
%T A197008 6,3,7,7,8,9,3,2,5,5,8,7,5,5,0,9,3,0,3,0,2,4,2,9,6,2,3,8,5,2,7,0,6,8,
%U A197008 8,5,0,3,6,5,0,2,9,1,5,9,3,8,2,4,6,1,3,8,8,2,2,0,6,7,8,3,6,1,2,3
%N A197008 Decimal expansion of the shortest distance from x axis through (1,2) to y axis.
%C A197008 The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P.  Suppose that T is the angle formed by the positive x and y axes and that h>0 and k>0.  Notation:
%C A197008 ...
%C A197008 P=(h,k)
%C A197008 L=the Philo line of P across T
%C A197008 U=x-intercept of L
%C A197008 V=y-intercept of L
%C A197008 d=|UV|
%C A197008 ...
%C A197008 Although Philo lines are not generally Euclidean-constructible, exact expressions for U, V, and d can be found for the angle T under consideration.  Write u(t)=(t,0), let v(t) the corresponding point on the y axis, and let d(t) be the distance between u(t) and v(t). Then d is found by minimizing d(t)^2:
%C A197008   d=w*sqrt(1+(k/h)^(2/3)), where w=(h+(h*k^2))^(1/3).
%C A197008 ...
%C A197008 Guide:
%C A197008 h....k...........d
%C A197008 1....2........A197008
%C A197008 1....3........A197012
%C A197008 1....4........A197013
%C A197008 2....3........A197014
%C A197008 3....4........A197015
%C A197008 1..sqrt(2)....A197031
%C A197008 ...
%C A197008 For guides to other Philo lines, see A195284 and A197032.
%C A197008 The cube root of any positive number can be connected to the Philo lines (or Philon lines) for a 90-degree angle. If the equation x^3-2 is represented using Lill's method, it can be shown that the path of the root 2^(1/3) creates the shortest segment (Philo line) from the x axis through (1,2) to the y axis. For more details see the article "Lill's method and the Philo Line for Right Angles" linked below. - _Raul Prisacariu_, Apr 06 2024
%H A197008 R. J. Mathar, <a href="/A197008/a197008.pdf">OEIS A197008</a>
%H A197008 Raul Prisacariu, <a href="https://web.archive.org/web/20231208023740/https://www.raulprisacariu.com/math/lills-method-and-the-philo-line-for-right-angles/">Lill's method and the Philo Line for Right Angles</a>.
%H A197008 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>
%e A197008 d=4.161938184941462752390080...
%e A197008 x-intercept: U=(2.5874..., 0)
%e A197008 y-intercept: V=(0, 3.2599...)
%p A197008 (1+2^(2/3))^(3/2); evalf(%) ; # _R. J. Mathar_, Nov 08 2022
%t A197008 f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);
%t A197008 h = 1; k = 2; d = N[f[t]^(1/2), 100]
%t A197008 RealDigits[d] (* this sequence *)
%t A197008 x = N[t] (* x-intercept; -1+4^(1/3); cf. A005480 *)
%t A197008 y = N[k*t/(t - h)] (* y-intercept *)
%t A197008 Show[Plot[k + k (x - h)/(h - t), {x, 0, t}],
%t A197008 ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 4}], PlotRange -> All, AspectRatio -> Automatic]
%o A197008 (PARI) polrootsreal(x^6 - 15*x^4 - 33*x^2 - 125)[2] \\ _Charles R Greathouse IV_, Feb 03 2025
%Y A197008 Cf. A197012, A005480, A195284.
%K A197008 nonn,cons
%O A197008 1,1
%A A197008 _Clark Kimberling_, Oct 10 2011