login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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