DarbouxFrenetSerret::usage="The Frenet-Serret frame applies to curves in space and refers to the trihedron of the tangent, normal and binormal vector to the curve. The Darboux frame is applied to a curve that is on a surface and we have a tangent vector to the curve, a normal vector to the surface and the normal tangent vector, orthogonal to the previous two. This animation shows both frames. Reference: Personal.";
In[]:=
Manipulateζ[t_]:=Cos[t]
2
Csc
t
2

,
2
Csc
t
2

Sin[t],2+Cos[t]
2
Csc
t
2

;σ[u_,v_]:={vCos[u],vSin[u],v};​​TF[t_]:=-
Csc
t
2
Sin[t]
2
(3+Cos[t])
,
2
2
(-1+Cos[t])
3+Cos[t]
3
Csc
t
2

,-
Csc
t
2
Sin[t]
2
(3+Cos[t])
;​​​​NF[t_]:=
3
Csc
t
2

4
Sin
t
2

3+Cos[t]
,-
3
Csc
t
2

2
Sin
t
2

Sin[t]
3+Cos[t]
,
3
Csc
t
2

4
Sin
t
2

3+Cos[t]
;​​BF[t_]:=-
6
Csc
t
2

6
Sin
t
2

2
,0,
6
Csc
t
2

6
Sin
t
2

2
;​​ND[t_]:=
Cos[t]
2
,
Sin[t]
2
,-
1
2
;​​TND[t_]:=-
(2+Cos[t])Sin
t
2

3+Cos[t]
,
2
3
Cos
t
2

3+Cos[t]
,
Sin
t
2

3+Cos[t]
;​​​​Show[{ParametricPlot3D[σ[u,v],{u,0,2π},{v,-12,12},PlotStyle->Directive[Yellow,Specularity[White,40],Opacity[0.4]],MeshFalse,PerformanceGoal->"Quality",PlotPoints60],ContourPlot3D[z-x==2,{x,-12,12},{y,-12,12},{z,-12,12},ContourStyle->{Blue,Opacity[0.1],Specularity[White,40]},MeshFalse],​​ParametricPlot3D[ζ[t],{t,0,t1},PlotStyle->{Black,Thick},PerformanceGoal->"Quality"],Graphics3D[​​{Black,Ball[ζ[t1],0.2],Point[ζ[t1]],​​​​Blue,Arrow[Tube[{ζ[t1],ζ[t1]+5N[TF[t1]]}]],​​Text[Style["
T
F
",14,Blue,Bold],ζ[t1]+5N[TF[t1]],{0,-1.5}],​​​​Red,Arrow[Tube[{ζ[t1],ζ[t1]+5N[NF[t1]]}]],​​Text[Style["
N
F
",14,Red,Bold],ζ[t1]+5N[NF[t1]],{0,-1.5}],​​​​Green,Arrow[Tube[{ζ[t1],ζ[t1]+5N[BF[t1]]}]],​​Text[Style["
B
F
",14,Green,Bold],ζ[t1]+5N[BF[t1]],{0,-1.5}],​​​​Brown,Arrow[Tube[{ζ[t1],ζ[t1]+5N[ND[t1]]}]],​​Text[Style["
N
D
",14,Brown,Bold],ζ[t1]+5N[ND[t1]],{0,-1.5}],​​​​Purple,Arrow[Tube[{ζ[t1],ζ[t1]+5N[TND[t1]]}]],​​Text[Style["
T
ND
",14,Purple,Bold],ζ[t1]+5N[TND[t1]],{0,-1.5}]​​}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange12,ViewPoint{6,-2.4,3.},ImageSize900],Style["Darboux and Frenet-Serret frames:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2π-0.00001,0.00001},ControlPlacementTop
Out[]=
​
Darboux and Frenet-Serret frames:
Value (t1)
Frenet-Serret Frame:
T
F
(blue),
N
F
(Red),
B
F
(Green).​Darboux Frame:
T
F
(blue),
N
D
(brown),
T
ND
(purple).