A332527 Decimal expansion of the maximal curvature of the tangent function.

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

Offset

0,1

Comments

The maximal curvature of the graph of y = tan x occurs at two points (x,y) on every branch. One of the points has y > 0. Let T be the branch passes through (0,0) and lies in the first quadrant. The maximal curvature, K, occurs at a point (u,v):

u = 0.69370020859538391768128598538590650878367123906075077978...

v = 0.83157590509648960702865222211498485994964124481665011305...

K = 0.37078258308108877400487185120239380769848079592957564055...

The osculating circle at (u,v) has

center = (x,y) = (-1.627936796879617446318318..., 2.204092389413177659055893...) .

radius = 1/K = 2.696998310142587559290309046607440826421048...

Links

Table of n, a(n) for n=0..85.

Example

maximal curvature: K = 0.370782583081088774004871851202393807698480795929575640...

Mathematica

centMin = {xMin = ArcCos[Root[3 - 4 #1^2 - 3 #1^4 + 2 #1^6 &, 3]],
   Root[-2 - 2 #1^2 + 5 #1^4 + 3 #1^6 &, 2]};
{centOsc, rOsc} = {{-(1/2) Cot[#1] (1 + Sec[#1]^4) + #1,
      Cot[#1] - 1/4 Sin[2 #1] + (3 Tan[#1])/2},
     Sqrt[1/4 Cos[#1]^4 Cot[#1]^2 (1 + Sec[#1]^4)^3]} &[xMin];
Show[Plot[{Tan[x], (-# Sec[#]^2) + x Sec[#]^2 +
      Tan[#], {(# Cos[#]^2) - x Cos[#]^2 + Tan[#]}}, {x, -5, 3},
    AspectRatio -> Automatic, ImageSize -> 500, PlotRange -> {-2, 4}],
    Graphics[{PointSize[Medium], Circle[centOsc, rOsc],
     Point[centOsc], Point[centMin]}]] &[xMin]
N[centOsc, 100]  (* center of osculating circle *)
N[rOsc, 100]  (* radius of osculating circle *)
N[{ArcCos[Root[3 - 4 #1^2 - 3 #1^4 + 2 #1^6 &, 3]],
  Root[-2 - 2 #1^2 + 5 #1^4 + 3 #1^6 &,
   2]}, 100] (* maximal curvature point *)
N[1/rOsc, 100]  (* curvature *)
(* Peter J. C. Moses, May 07 2020 *)

Crossrefs

Cf. A332527.

Sequence in context: A199778 A369381 A086729 * A175576 A134976 A192044

Adjacent sequences: A332524 A332525 A332526 * A332528 A332529 A332530

Keyword

nonn,cons

Author

Clark Kimberling, Jun 15 2020

Status

approved