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A332528
Decimal expansion of the maximal curvature of the secant function.
0
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...
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
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jun 21 2020
STATUS
approved