A197153 -id:A197153 - cp's OEIS frontend

A197032 Decimal expansion of the x-intercept of the shortest segment from the positive x axis through (2,1) to the line y=x.

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

Offset: 1

Clark Kimberling, Oct 10 2011

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
1......2......1.......A197032......A197033
1......3......1.......A197034......A197035
1......4......1.......A197136......A197137
1......3......2.......A197138......A197139
2......1......1.......A197140......A197141
2......2......1.......A197142......A197143
2......3......1.......A197144......A197145
2......4......1.......A197146......A197147
3......1......1.......A197148......A197149
3......2......1.......A197150......A197151
1/2....3......1.......A197152......A197153
1/2....4......1.......A197154......A197155

			length of Philo line:  1.8442716817001... (see A197033)
endpoint on x axis: (2.35321..., 0)
endpoint on y=x:    (1.73898, 1.73898)

R. J. Mathar, OEIS A197032, Nov. 8, 2022
M. F. Hasler, Philo line - oeis.org/A197032 (google drawing), Nov. 8, 2022
Wikipedia, Philo line
Index entries for algebraic numbers, degree 3

Cf. A197033, A197008, A195284.

Cf. A357469 (= this constant - 1).

Digits := 140 ;
x^3-4*x^2+6*x-5 ;
fsolve(%=0) ; # R. J. Mathar, Nov 08 2022

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]
RealDigits[t]  (* A197032 *)
{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]
RealDigits[d] (* A197033 *)
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]

PARI
```
solve(x=2,3, x^3 - 4*x^2 + 6*x - 5)
```

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

Invalid trailing digits corrected by R. J. Mathar, Nov 08 2022

A197152 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 11 2011

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.

A root of the polynomial x^3/2 -5*x^2/2 +9*x/2 -5. - R. J. Mathar, Nov 08 2022

			length of Philo line:    1.481506505...; see A197153
endpoint on x axis:    (3.15091, 0)
endpoint on line y=3x: (2.92984, 1.46492)

Cf. A197032, A197153, A197008, A195284.

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 = 1/2; h = 3; k = 1;(* slope m, point (h,k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t] (* A197152 *)
{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=x/2 *)
d = N[Sqrt[f[t]], 100]
RealDigits[d]  (* A197153 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3.5}],
ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}],
PlotRange -> {0, 1.5}, AspectRatio -> Automatic]

Incorrect trailing digits deleted. - R. J. Mathar, Nov 08 2022

cp's OEIS Frontend

A197032 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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