A197148 Decimal expansion of the x-intercept of the shortest segment from the x axis through (1,1) to the line y=3x.

1, 6, 0, 4, 7, 9, 3, 6, 1, 8, 4, 6, 2, 1, 3, 9, 9, 0, 7, 3, 7, 8, 3, 1, 7, 9, 5, 0, 7, 1, 7, 9, 6, 1, 8, 4, 6, 7, 1, 5, 4, 4, 9, 2, 1, 9, 9, 9, 1, 2, 8, 6, 0, 7, 7, 8, 6, 3, 6, 2, 9, 2, 2, 1, 4, 9, 2, 1, 6, 3, 7, 2, 6, 1, 9, 1, 2, 6, 0, 4, 2, 1, 6, 6, 7, 9, 9, 7, 0, 2, 2, 8, 4, 7, 0, 1, 4, 7, 7, 2

Offset: 1

Author

Clark Kimberling, Oct 11 2011

Keywords

• nonn
• cons

Comments

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.

Examples

			length of Philo line:  1.999158399580...; see A197149
endpoint on x axis:    (1.60479, 0)
endpoint on line y=3x: (0.570212, 1.71064)
		

Crossrefs

Cf. A197032, A197149, A197008, A195284.

Programs

Mathematica

f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
g[t_] := D[f[t], t]; Factor[g[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
m = 3; h = 1; k = 1;(* slope m, point (h,k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t] (* A197148 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=3x *)
d = N[Sqrt[f[t]], 100]
RealDigits[d]  (* A197149 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]