DarbouxVector::usage="The Darboux vector of a space curve with unit speed is the angular velocity vector of the Frenet-Serret frame. And it is calculated like this: D = τ T + κ B. Where τ is the torsion and κ the curvature. Reference: DarbouxVector, FunctionRepository; Personal. ";
In[]:=
Manipulateζ[t_]:=2(1+Cos[t]),2Sin[t],4Sin
t
2
;​​T[t_]:=-
2
Sin[t]
3+Cos[t]
,
2
Cos[t]
3+Cos[t]
,
2
Cos
t
2

3+Cos[t]
;​​​​N1[t_]:=-
3+12Cos[t]+Cos[2t]
2
3+Cos[t]
13+3Cos[t]
,-
(6+Cos[t])Sin[t]
3+Cos[t]
13+3Cos[t]
,-
2Sin
t
2

3+Cos[t]
13+3Cos[t]
;​​B[t_]:=
2(2+Cos[t])Sin
t
2

26+6Cos[t]
,-
2
2
3
Cos
t
2

13+3Cos[t]
,
2
2
13+3Cos[t]
;​​​​Dar[t_]:=
2(8+3Cos[t])
3
Sin
t
2

(3+Cos[t])(13+3Cos[t])
,
Cos
t
2
(-23+4Cos[t]+3Cos[2t])
2(3+Cos[t])(13+3Cos[t])
,1+
2
3+Cos[t]
-
10
13+3Cos[t]
;​​​​Show[{​​ParametricPlot3D[ζ[t],{t,0,t1},PlotStyle->{Black,Thick}],Graphics3D[​​{Black,Ball[ζ[t1],0.16],Point[ζ[t1]],​​​​Blue,Arrow[Tube[{ζ[t1],ζ[t1]+8N[T[t1]]}]],​​Text[Style["T",14,Blue,Bold],ζ[t1]+8N[T[t1]],{0,-1.5}],​​​​Red,Arrow[Tube[{ζ[t1],ζ[t1]+8N[N1[t1]]}]],​​Text[Style["N",14,Red,Bold],ζ[t1]+8N[N1[t1]],{0,-1.5}],​​​​Green,Arrow[Tube[{ζ[t1],ζ[t1]+8N[B[t1]]}]],​​Text[Style["B",14,Green,Bold],ζ[t1]+8N[B[t1]],{0,-1.5}],​​​​Brown,Arrow[Tube[{ζ[t1],ζ[t1]+8N[Dar[t1]]}]],​​Text[Style["D",14,Brown,Bold],ζ[t1]+8N[Dar[t1]],{0,-1.5}]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-10,8},{-10,5},{-10,12}},ViewPoint{1.3,-2.4,2.},ImageSize700],Style["Darboux vector and Frenet-Serret frame:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,4π,0.00001},ControlPlacementTop
Out[]=
​
Darboux vector and Frenet-Serret frame:
Value (t1)
In[]:=
ManipulateDar[t_]:=
2(8+3Cos[t])
3
Sin
t
2

(3+Cos[t])(13+3Cos[t])
,
Cos
t
2
(-23+4Cos[t]+3Cos[2t])
2(3+Cos[t])(13+3Cos[t])
,1+
2
3+Cos[t]
-
10
13+3Cos[t]
;​​​​p=Grid[{{DarbouxVector},{Dar[t1]}},FrameAll,ItemSize->15,ItemStyle->15];​​Show[{​​ParametricPlot3D[Dar[t],{t,0,t1},PlotStyle->{Brown,Thick}],Graphics3D[​​{Brown,Ball[Dar[t1],0.01]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-0.6,0.6},{-0.6,0.6},{-0.1,1}},ViewPoint{1.3,-2.4,2.},ImageSize700],Style["Darboux vector:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,4π,0.00001},Delimiter,{{p,2,"Darboux vector coordinates"}},ControlPlacementTop
Out[]=
​
Darboux vector:
Value (t1)
Darboux vector coordinates
DarbouxVector
-9.96368×
-21
10
,-0.125,0.875
Darboux components graph