A332526 Decimal expansion of the minimal distance between distinct branches of the tangent function; see Comments.

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

Offset: 1

Author

Clark Kimberling, Jun 15 2020

Keywords

• nonn
• cons

Comments

Let T0 and T1 be the branches of the graph of y = tan x that passes through (0,0,) and (Pi,0), respectively. There exist points P = (p,q) on T0 and U = (u,v) on T1 such that the distance between P and U is the minimal distance, d, between points on T0 and T1.

u = 2.549082584017596768984130292562154758705824602711...

v = -0.67319711901285205370684801604861382107848678888...

p = Pi - u

q = - v

d = 2.375069146040176349439851558778982487866267806508...

Examples

			minimal distance = 2.375069146040176349439851558778982487866267806508...
		

Crossrefs

Cf. A332500, A332525, A332527.

Programs

Mathematica

min = Quiet[FindMinimum[Sqrt[(#[[1]][[1]] - #[[2]][[1]])^2 + (#[[1]][[2]] - \
#[[2]][[2]])^2] &[{{#, Tan[#]} &[x /. FindRoot[# Cos[#]^2 - x Cos[#]^2 + Tan[#] == Tan[x], {x, 0}, WorkingPrecision -> 500]], {#, Tan[#]} &[#]} &[y]], {y, 2}, WorkingPrecision -> 100]]
Show[Plot[{Tan[x], (-# Sec[#]^2) + x Sec[#]^2 + Tan[#], {(# Cos[#]^2) - x Cos[#]^2 + Tan[#]}}, {x, 0, Pi}, AspectRatio -> Automatic, ImageSize -> 300, PlotRange -> {-2, 2}], Graphics[{PointSize[Large], Point[{Pi/2, 0}], Point[{#, Tan[#]}], Point[{Pi - #, -Tan[#]}]}]] &[y /. min[[2]][[1]]]
(* Peter J. C. Moses, May 06 2020 *)