A197138 -id:A197138 - 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

A197139 Decimal expansion of the shortest distance from the x axis through (3,2) to the line y = x.

Original entry on oeis.org

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

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 A197032, A197008 and A195284.

Any Philo line can be constructed using the intersections of circles and hyperbolas. In this case, the Philo line passes though the two points at which the circle (x - 3/2)^2 + (y - 1)^2 = 13/4 and the hyperbola x*y - y^2 = 2 intersect. The circle has segment OP as diameter, where O(0,0) is the origin and P is the point (3,2). The asymptotes of the hyperbola are the x axis and the line y = x. Point P is one of the two points at which the circle and the hyperbola intersect. - Raul Prisacariu, Apr 06 2024

			Length of Philo line:   2.8861171058980012915367...
Endpoint on x axis:     (3.4883, 0); see A197138
Endpoint on line y = x: (2.80376, 2.80376)

Cf. A197032, A197138, 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; h = 3; k = 2;(* slope m; point (h,k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A197138 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* endpoint on line y=x *)
d = N[Sqrt[f[t]], 100]
RealDigits[d]  (* this sequence *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 4}, {y, 0, 3}],
PlotRange -> {0, 3}, AspectRatio -> Automatic]

Last digit removed (repr. truncated, not rounded up) by 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions