A197031 -id:A197031 - cp's OEIS frontend

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, 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, 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

Offset: 1

Clark Kimberling, Oct 10 2011

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:
...
P=(h,k)
L=the Philo line of P across T
U=x-intercept of L
V=y-intercept of L
d=|UV|
...
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:
  d=w*sqrt(1+(k/h)^(2/3)), where w=(h+(h*k^2))^(1/3).
...
Guide:
h....k...........d
1....2........A197008
1....3........A197012
1....4........A197013
2....3........A197014
3....4........A197015
1..sqrt(2)....A197031
...
For guides to other Philo lines, see A195284 and A197032.
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

			d=4.161938184941462752390080...
x-intercept: U=(2.5874..., 0)
y-intercept: V=(0, 3.2599...)

R. J. Mathar, OEIS A197008
Raul Prisacariu, Lill's method and the Philo Line for Right Angles.
Index entries for algebraic numbers, degree 6

Cf. A197012, A005480, A195284.

(1+2^(2/3))^(3/2); evalf(%) ; # R. J. Mathar, Nov 08 2022

f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);
h = 1; k = 2; d = N[f[t]^(1/2), 100]
RealDigits[d] (* this sequence *)
x = N[t] (* x-intercept; -1+4^(1/3); cf. A005480 *)
y = N[k*t/(t - h)] (* y-intercept *)
Show[Plot[k + k (x - h)/(h - t), {x, 0, t}],
ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 4}], PlotRange -> All, AspectRatio -> Automatic]

polrootsreal(x^6 - 15*x^4 - 33*x^2 - 125)[2] \\ Charles R Greathouse IV, Feb 03 2025

cp's OEIS Frontend

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

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