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A332527 Decimal expansion of the maximal curvature of the tangent function. 4
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 (list; constant; graph; refs; listen; history; text; internal format)
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: A197005 A199778 A086729 * A175576 A134976 A192044

Adjacent sequences:  A332524 A332525 A332526 * A332528 A332529 A332530

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jun 15 2020

STATUS

approved

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Last modified April 19 15:00 EDT 2021. Contains 343116 sequences. (Running on oeis4.)