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A197008
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Decimal expansion of the shortest distance from x axis through (1,2) to y axis.
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32
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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
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OFFSET
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1,1
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COMMENTS
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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:
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P=(h,k)
L=the Philo line of P across T
U=x-intercept of L
V=y-intercept of L
d=|UV|
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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).
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Guide:
h....k...........d
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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
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LINKS
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EXAMPLE
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d=4.161938184941462752390080...
x-intercept: U=(2.5874..., 0)
y-intercept: V=(0, 3.2599...)
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MAPLE
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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