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

Author

Clark Kimberling, Jun 21 2020

Keywords

• nonn
• cons

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...

Crossrefs

Cf. A332527.

Programs

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 *)