OFFSET
1,1
COMMENTS
LINKS
R. J. Mathar, OEIS A197032, Nov. 8, 2022
M. F. Hasler, Philo line - oeis.org/A197032 (google drawing), Nov. 8, 2022
Wikipedia, Philo line
FORMULA
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
EXAMPLE
length of Philo line: 1.8442716817001... (see A197033)
endpoint on x axis: (2.35321..., 0)
endpoint on y=x: (1.73898, 1.73898)
MAPLE
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 (* 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]
PROG
(PARI) solve(x=2, 3, x^3 - 4*x^2 + 6*x - 5)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 10 2011
EXTENSIONS
Invalid trailing digits corrected by R. J. Mathar, Nov 08 2022
STATUS
approved