A197035 - OEIS

%I #9 Mar 30 2012 18:57:52

%S 2,6,0,8,1,9,4,0,2,4,9,6,1,0,1,8,9,0,1,9,9,0,1,4,4,5,4,2,8,3,5,2,2,3,

%T 9,5,9,0,8,3,5,8,9,1,1,5,8,7,9,5,9,7,6,7,4,4,9,4,9,1,7,7,5,6,3,8,5,7,

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

%N Decimal expansion of the shortest distance from the x axis through (3,1) to the line y=x.

%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 A197008 and A195284.

%e length of Philo line:  2.60819402496101...

%e endpoint on x axis:   (3.47797, 0); see A197034

%e endpoint on line y=x: (2.35321, 2.35321)

%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 = 1; h = 3; k = 1;  (* slope m; point (h,k) *)

%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]

%t RealDigits[t]  (* A197034 *)

%t {N[t], 0} (* endpoint on x axis *)

%t {N[k*t/(k + m*t - m*h)], N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)

%t d = N[Sqrt[f[t]], 100]

%t RealDigits[d] (* A197035 *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],

%t ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 3.5}, {y, 0, 3}], PlotRange -> {0, 3}, AspectRatio -> Automatic]

%Y Cf. A197032, A197034, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 10 2011