|
%I #7 Nov 08 2022 11:41:23
%S 1,9,9,9,1,5,8,3,9,9,5,8,0,3,4,4,2,6,8,8,1,7,4,2,3,5,4,3,8,4,6,1,6,4,
%T 7,3,4,1,3,2,8,2,3,4,7,1,1,8,9,0,6,8,3,7,7,0,9,2,8,1,2,8,1,0,4,6,8,5,
%U 4,8,5,2,8,7,0,0,4,9,0,3,2,7,6,5,3,7,3,7,5,1,3,1,8,9,2,0,5,5,5
%N Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=3x.
%C The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T. For discussions and guides to related sequences, see A197032, A197008 and A195284.
%e length of Philo line: 1.999158399580...
%e endpoint on x axis: (1.60479, 0); see A197148
%e endpoint on line y=3x: (0.570212, 1.71064)
%t f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t g[t_] := D[f[t], t]; Factor[g[t]]
%t p[t_] := h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3
%t m = 3; h = 1; k = 1;(* slope m, point (h,k) *)
%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t RealDigits[t] (* A197148 *)
%t {N[t], 0} (* endpoint on x axis *)
%t {N[k*t/(k + m*t - m*h)],
%t N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=3x *)
%t d = N[Sqrt[f[t]], 100]
%t RealDigits[d] (* A197149 *)
%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
%t ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]
%Y Cf. A197032, A197049, A197008, A195284.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 11 2011
|
