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%I #8 Jun 01 2022 12:16:45
%S 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,
%T 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,
%U 3,4,6,7,6,8,6,4,7,3,9,2,1,7,4,3,2,3
%N Decimal expansion of the maximal curvature of the secant function.
%C 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):
%C u = 0.469952511643664772466732023628843853062603014858623133147...
%C v = 1.121592022152314185447110000884194699579672726085862403985...
%C K = 1.11539861636708478852367202281326657654751208184531773740...
%C The osculating circle at (u,v) has
%C center = (x,y) = (0.02618081309772817465,,,, 1.900598757881329358432040976889617...).
%C radius = 1/K = 0.896540470219566446984489512566284091376257661443...
%C maximal curvature: K = 1.11539861636708478852367202281326657654751208184531773740...
%t maxC = ArcCos[ Sqrt[(2*(2 - Sqrt[31]*Cos[(Pi + ArcTan[(9*Sqrt[302])/73])/3]))/3]];
%t centerMaxC = {(-3*x + x*Cos[2*x] + Sin[2*x] + 2*Tan[x]^3)/(-3 +
%t Cos[2*x]), -((Cos[x]*Cos[2*x])/(-3 + Cos[2*x])) + (3*Sec[x])/
%t 2} /. x -> maxC;
%t rMin = (Sqrt[(7 + Cos[4 x])^3] Sec[x]^3)/(Sqrt[2] 8 (3 - Cos[2 x])) /. x -> maxC;
%t Show[Plot[Sec[x], {x, 0 - 3/2, 2}],
%t Graphics[{PointSize[Large], Red, Point[centerMaxC],
%t Point[{maxC, Sec[maxC]}], Circle[centerMaxC, rMin],
%t Line[{centerMaxC, {maxC, Sec[maxC]}}]}], AspectRatio -> Automatic,
%t PlotRange -> {0, 2}]
%t x = ArcCos[Sqrt[(2*(2 - Sqrt[31]*Cos[(Pi + ArcTan[(9*Sqrt[302])/73])/3]))/
%t 3]]; (* maximal curvature occurs at (x, sec x) *)
%t {N[x, 150], N[Sec[x], 150]}
%t {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 *)
%t {N[cx, 150], N[cy, 150]}
%t r = N[Sqrt[(x - cx)^2 + (Sec[x] - cy)^2], 150] (* radius of curvature *)
%t 1/r (* curvature *)
%t kr = RealDigits[1/r][[1]]
%t (* _Peter J. C. Moses_, May 07 2020 *)
%Y Cf. A332527.
%K nonn,cons
%O 0,4
%A _Clark Kimberling_, Jun 21 2020
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