login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A332527 Decimal expansion of the maximal curvature of the tangent function. 4
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, 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, 4, 2, 7, 6, 5, 8, 3, 6, 5, 5, 4, 7, 8, 5, 1, 6, 5, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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):
u = 0.69370020859538391768128598538590650878367123906075077978...
v = 0.83157590509648960702865222211498485994964124481665011305...
K = 0.37078258308108877400487185120239380769848079592957564055...
The osculating circle at (u,v) has
center = (x,y) = (-1.627936796879617446318318..., 2.204092389413177659055893...) .
radius = 1/K = 2.696998310142587559290309046607440826421048...
LINKS
EXAMPLE
maximal curvature: K = 0.370782583081088774004871851202393807698480795929575640...
MATHEMATICA
centMin = {xMin = ArcCos[Root[3 - 4 #1^2 - 3 #1^4 + 2 #1^6 &, 3]],
Root[-2 - 2 #1^2 + 5 #1^4 + 3 #1^6 &, 2]};
{centOsc, rOsc} = {{-(1/2) Cot[#1] (1 + Sec[#1]^4) + #1,
Cot[#1] - 1/4 Sin[2 #1] + (3 Tan[#1])/2},
Sqrt[1/4 Cos[#1]^4 Cot[#1]^2 (1 + Sec[#1]^4)^3]} &[xMin];
Show[Plot[{Tan[x], (-# Sec[#]^2) + x Sec[#]^2 +
Tan[#], {(# Cos[#]^2) - x Cos[#]^2 + Tan[#]}}, {x, -5, 3},
AspectRatio -> Automatic, ImageSize -> 500, PlotRange -> {-2, 4}],
Graphics[{PointSize[Medium], Circle[centOsc, rOsc],
Point[centOsc], Point[centMin]}]] &[xMin]
N[centOsc, 100] (* center of osculating circle *)
N[rOsc, 100] (* radius of osculating circle *)
N[{ArcCos[Root[3 - 4 #1^2 - 3 #1^4 + 2 #1^6 &, 3]],
Root[-2 - 2 #1^2 + 5 #1^4 + 3 #1^6 &,
2]}, 100] (* maximal curvature point *)
N[1/rOsc, 100] (* curvature *)
(* Peter J. C. Moses, May 07 2020 *)
CROSSREFS
Cf. A332527.
Sequence in context: A199778 A369381 A086729 * A175576 A134976 A192044
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jun 15 2020
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 06:15 EDT 2024. Contains 371265 sequences. (Running on oeis4.)