In[1]:=

t0 = 17 ; r = {t/7, Cos[t], Sin[t]} ; T = D[r, t]/(D[r, t] . D[r, t])^(1/2)//Simplify ; κ ... irc = ParametricPlot3D[circ0, {θ, 0, 2π}, PlotPoints500] ; Show[helix, osccirc]

r =  {t/7, Cos[t], Sin[t]}

r' =  {1/7, -Sin[t], Cos[t]}

r'' =  {0, -Cos[t], -Sin[t]}

unit Tangent vector T =  {1/(5 2^(1/2)), -(7 Sin[t])/(5 2^(1/2)), (7 Cos[t])/(5 2^(1/2))}

unit normal vector N =  {0, -Cos[t], -Sin[t]}

3 curvature κ = |r' x r''|/|r' | = 49/50

radius of curvature ρ = 50/49

At t = 19 :

r  is  {19/7, Cos[19], Sin[19]}

T  is  {1/(5 2^(1/2)), -(7 Sin[19])/(5 2^(1/2)), (7 Cos[19])/(5 2^(1/2))}

N is  {0, -Cos[19], -Sin[19]}

ρ is 50/49

osculating circle is  {19/7 + 5/49 2^(1/2) Cos[θ], Cos[19] + 50/49 (-Cos[19] - (7 ... s[θ])/(5 2^(1/2)) - Sin[19] - Sin[19] Sin[θ])}  for 0≤θ≤2π

[Graphics:HTMLFiles/osculating-circle_15.gif]

Out[22]=

⁃Graphics3D⁃

Created by Mathematica  (February 11, 2013)