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

Author

Clark Kimberling, Jun 15 2020

Keywords

• nonn
• cons

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...

Examples

			maximal curvature: K = 0.370782583081088774004871851202393807698480795929575640...
		

Crossrefs

Cf. A332527.

Programs

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 *)