Search: a197140 -id:a197140
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A197032
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Decimal expansion of the x-intercept of the shortest segment from the positive x axis through (2,1) to the line y=x.
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+10
26
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2, 3, 5, 3, 2, 0, 9, 9, 6, 4, 1, 9, 9, 3, 2, 4, 4, 2, 9, 4, 8, 3, 1, 0, 1, 3, 3, 2, 5, 7, 7, 3, 8, 8, 4, 5, 7, 2, 7, 0, 7, 0, 5, 6, 1, 3, 8, 5, 6, 8, 4, 6, 8, 2, 6, 8, 0, 6, 6, 9, 3, 0, 4, 2, 6, 5, 1, 5, 1, 8, 9, 7, 2, 3, 2, 2, 0, 9, 2, 0, 8, 5, 9, 1, 6, 5, 8, 0, 3
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OFFSET
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1,1
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COMMENTS
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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.
Philo lines from positive x axis through (h,k) to line y=mx:
m......h......k....x-intercept.....distance
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LINKS
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FORMULA
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x = 2 + tan phi where 1 + 2 tan phi = 1/(sin phi + cos phi), whence x = 1 + A357469 = the only real root of x^3 - 4*x^2 + 6*x - 5. - M. F. Hasler, Nov 08 2022
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EXAMPLE
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length of Philo line: 1.8442716817001... (see A197033)
endpoint on x axis: (2.35321..., 0)
endpoint on y=x: (1.73898, 1.73898)
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MAPLE
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Digits := 140 ;
x^3-4*x^2+6*x-5 ;
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MATHEMATICA
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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 (* root of p[t] minimizes f *)
m = 1; h = 2; k = 1; (* m=slope; (h, k)=point *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
{N[t], 0} (* lower endpoint of minimal segment [Philo line] *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
d = N[Sqrt[f[t]], 100]
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2.5}],
ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 3}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]
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PROG
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(PARI) solve(x=2, 3, x^3 - 4*x^2 + 6*x - 5)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Invalid trailing digits corrected by R. J. Mathar, Nov 08 2022
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STATUS
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approved
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A197141
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Decimal expansion of the shortest distance from the x axis through (1,1) to the line y=2x.
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+10
3
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1, 6, 7, 3, 6, 4, 7, 3, 0, 4, 1, 5, 2, 9, 1, 5, 0, 7, 8, 0, 1, 3, 8, 6, 3, 4, 3, 3, 2, 7, 8, 1, 6, 6, 0, 2, 6, 8, 5, 8, 3, 6, 5, 7, 7, 1, 0, 3, 5, 3, 9, 2, 8, 6, 1, 7, 9, 9, 4, 6, 0, 5, 6, 9, 5, 2, 6, 1, 8, 9, 5, 6, 2, 8, 0, 5, 4, 7, 5, 7, 2, 9, 1, 1, 9, 3, 7, 1, 7, 0, 9, 5, 8, 5, 1, 2, 9, 5, 3
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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length of Philo line: 1.6736473041529...
endpoint on x axis: (1.44062, 0); see A197140
endpoint on line y=2x: (0.765782, 1.53156)
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MATHEMATICA
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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 = 2; h = 1; k = 1; (* slope m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
{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=mx *)
d = N[Sqrt[f[t]], 100]
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, 1.7}, 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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