A332528 Decimal expansion of the maximal curvature of the secant function.

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

Offset

0,4

Comments

The maximal curvature of the graph of y = sec x occurs at two points (x,y) on every branch. One of the points has y > 0. Let T be the branch passes through (0,1) and lies in the first quadrant. The maximal curvature, K, occurs at a point (u,v):

u = 0.469952511643664772466732023628843853062603014858623133147...

v = 1.121592022152314185447110000884194699579672726085862403985...

K = 1.11539861636708478852367202281326657654751208184531773740...

The osculating circle at (u,v) has

center = (x,y) = (0.02618081309772817465,,,, 1.900598757881329358432040976889617...).

radius = 1/K = 0.896540470219566446984489512566284091376257661443...

maximal curvature: K = 1.11539861636708478852367202281326657654751208184531773740...

Links

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

Mathematica

maxC = ArcCos[   Sqrt[(2*(2 - Sqrt[31]*Cos[(Pi + ArcTan[(9*Sqrt[302])/73])/3]))/3]];
centerMaxC = {(-3*x + x*Cos[2*x] + Sin[2*x] + 2*Tan[x]^3)/(-3 +
       Cos[2*x]), -((Cos[x]*Cos[2*x])/(-3 + Cos[2*x])) + (3*Sec[x])/
      2} /. x -> maxC;
rMin = (Sqrt[(7 + Cos[4 x])^3] Sec[x]^3)/(Sqrt[2] 8 (3 - Cos[2 x])) /. x -> maxC;
Show[Plot[Sec[x], {x, 0 - 3/2, 2}],
Graphics[{PointSize[Large], Red, Point[centerMaxC],
   Point[{maxC, Sec[maxC]}], Circle[centerMaxC, rMin],
   Line[{centerMaxC, {maxC, Sec[maxC]}}]}], AspectRatio -> Automatic,
PlotRange -> {0, 2}]
x = ArcCos[Sqrt[(2*(2 - Sqrt[31]*Cos[(Pi + ArcTan[(9*Sqrt[302])/73])/3]))/
    3]];  (* maximal curvature occurs at (x, sec x) *)
{N[x, 150], N[Sec[x], 150]}
{cx, cy} = {(-3*x + x*Cos[2*x] + Sin[2*x] + 2*Tan[x]^3)/(-3 + Cos[2*x]), -((Cos[x]*Cos[2*x])/(-3 + Cos[2*x])) + (3*Sec[x])/2}; (* center of osculating circle *)
{N[cx, 150], N[cy, 150]}
r = N[Sqrt[(x - cx)^2 + (Sec[x] - cy)^2], 150] (* radius of curvature *)
1/r  (* curvature *)
kr = RealDigits[1/r][[1]]
(* Peter J. C. Moses, May 07 2020 *)

Crossrefs

Cf. A332527.

Sequence in context: A224511 A201938 A201410 * A019955 A296345 A317907

Adjacent sequences: A332525 A332526 A332527 * A332529 A332530 A332531

Keyword

nonn,cons

Author

Clark Kimberling, Jun 21 2020

Status

approved