cp's OEIS Frontend

A332527 Decimal expansion of the maximal curvature of the tangent function.

%I A332527 #8 Jun 01 2022 12:16:31
%S A332527 3,7,0,7,8,2,5,8,3,0,8,1,0,8,8,7,7,4,0,0,4,8,7,1,8,5,1,2,0,2,3,9,3,8,
%T A332527 0,7,6,9,8,4,8,0,7,9,5,9,2,9,5,7,5,6,4,0,5,5,7,3,9,3,3,0,3,0,3,4,1,3,
%U A332527 4,2,7,6,5,8,3,6,5,5,4,7,8,5,1,6,5,1
%N A332527 Decimal expansion of the maximal curvature of the tangent function.
%C A332527 The maximal curvature of the graph of y = tan x occurs at two points (x,y) on every branch.  One of the points has y > 0. Let T be the branch passes through (0,0) and lies in the first quadrant. The maximal curvature, K, occurs at a point (u,v):
%C A332527 u = 0.69370020859538391768128598538590650878367123906075077978...
%C A332527 v = 0.83157590509648960702865222211498485994964124481665011305...
%C A332527 K = 0.37078258308108877400487185120239380769848079592957564055...
%C A332527 The osculating circle at (u,v) has
%C A332527 center = (x,y) = (-1.627936796879617446318318..., 2.204092389413177659055893...) .
%C A332527 radius = 1/K = 2.696998310142587559290309046607440826421048...
%e A332527 maximal curvature: K = 0.370782583081088774004871851202393807698480795929575640...
%t A332527 centMin = {xMin = ArcCos[Root[3 - 4 #1^2 - 3 #1^4 + 2 #1^6 &, 3]],
%t A332527    Root[-2 - 2 #1^2 + 5 #1^4 + 3 #1^6 &, 2]};
%t A332527 {centOsc, rOsc} = {{-(1/2) Cot[#1] (1 + Sec[#1]^4) + #1,
%t A332527       Cot[#1] - 1/4 Sin[2 #1] + (3 Tan[#1])/2},
%t A332527      Sqrt[1/4 Cos[#1]^4 Cot[#1]^2 (1 + Sec[#1]^4)^3]} &[xMin];
%t A332527 Show[Plot[{Tan[x], (-# Sec[#]^2) + x Sec[#]^2 +
%t A332527       Tan[#], {(# Cos[#]^2) - x Cos[#]^2 + Tan[#]}}, {x, -5, 3},
%t A332527     AspectRatio -> Automatic, ImageSize -> 500, PlotRange -> {-2, 4}],
%t A332527     Graphics[{PointSize[Medium], Circle[centOsc, rOsc],
%t A332527      Point[centOsc], Point[centMin]}]] &[xMin]
%t A332527 N[centOsc, 100]  (* center of osculating circle *)
%t A332527 N[rOsc, 100]  (* radius of osculating circle *)
%t A332527 N[{ArcCos[Root[3 - 4 #1^2 - 3 #1^4 + 2 #1^6 &, 3]],
%t A332527   Root[-2 - 2 #1^2 + 5 #1^4 + 3 #1^6 &,
%t A332527    2]}, 100] (* maximal curvature point *)
%t A332527 N[1/rOsc, 100]  (* curvature *)
%t A332527 (* _Peter J. C. Moses_, May 07 2020 *)
%Y A332527 Cf. A332527.
%K A332527 nonn,cons
%O A332527 0,1
%A A332527 _Clark Kimberling_, Jun 15 2020