SeveralAspectsHelix::usage="Demonstrate that the tangents to the helixx = a Cos[ω t], y = a Sin[ω t], z = b tform a constant angle with the XY plane. Demonstrate that the main normalsof the helix cut off to the Z axis. Prove that the curvature and torsion areconstant.References:Pogorélov,A.V.GeometríaDiferencial.Moscú:Ed.Mir:p.84,1977;Personal.";
Development:
In[]:=
ζ[t_]:={aCos[ωt],aSin[ωt],bt};Cos[θ]==
{0,0,1}.ζ'[t]
Norm[{0,0,1}].Norm[ζ'[t]]
In[]:=
SimplifyCos[θ]
b
1.++
2
b
2
(aωCos[tω])
2
(aωSin[tω])
Out[]=
Cos[θ]
b
1.+
2
b
2
a
2
ω
The angle formed by the tangents of the helix with the plane XY is constant..
In[]:=
ResourceFunction["NormalVector"][ζ[t],t]
In[]:=
Simplify-(+)Cos[tω]--,-(+)Sin[tω]--,0
2
b
2
a
4
ω
2
b
2
a
2
ω
a
2
ω
3/2
(+)
2
b
2
a
2
ω
a(+)
2
a
4
ω
2
b
2
a
2
ω
3
Cos[tω]
3/2
(+)
2
b
2
a
2
ω
a(+)Cos[tω]
2
a
4
ω
2
b
2
a
2
ω
2
Sin[tω]
3/2
(+)
2
b
2
a
2
ω
2
b
2
a
4
ω
2
b
2
a
2
ω
a
2
ω
3/2
(+)
2
b
2
a
2
ω
a(+)Sin[tω]
2
a
4
ω
2
b
2
a
2
ω
2
Cos[tω]
3/2
(+)
2
b
2
a
2
ω
a(+)
2
a
4
ω
2
b
2
a
2
ω
3
Sin[tω]
3/2
(+)
2
b
2
a
2
ω
Out[]=
-Cos[tω],-Sin[tω],0
3/2
((+))
2
a
4
ω
2
b
2
a
2
ω
3
a
6
ω
3/2
(+)
2
b
2
a
2
ω
3/2
((+))
2
a
4
ω
2
b
2
a
2
ω
3
a
6
ω
3/2
(+)
2
b
2
a
2
ω
In[]:=
-Cos[tω],-Sin[tω],0.{0,0,1}
3/2
((+))
2
a
4
ω
2
b
2
a
2
ω
3
a
6
ω
3/2
(+)
2
b
2
a
2
ω
3/2
((+))
2
a
4
ω
2
b
2
a
2
ω
3
a
6
ω
3/2
(+)
2
b
2
a
2
ω
Out[]=
0
They are orthogonal, so the main normals of the helix cut to the Z axis.
In[]:=
ResourceFunction["Curvature"][ζ[t],t]
In[]:=
Simplify++
2
a
2
b
4
ω
2
Cos[tω]
2
a
2
b
4
ω
2
Sin[tω]
2
(+)
2
a
3
ω
2
Cos[tω]
2
a
3
ω
2
Sin[tω]
3/2
(++)
2
b
2
a
2
ω
2
Cos[tω]
2
a
2
ω
2
Sin[tω]
2
a
4
ω
2
b
2
a
2
ω
3/2
(+)
2
b
2
a
2
ω
a
2
ω
2
b
2
a
2
ω
The curvature of the helix is constant.
In[]:=
χ==
Det[{ζ'[t],ζ''[t],ζ'''[t]}]
2
Norm[ζ'[t]ζ''[t]]
In[]:=
Simplifyb+b++
2
a
5
ω
2
Cos[tω]
2
a
5
ω
2
Sin[tω]
2
(abCos[tω])
2
ω
2
(abSin[tω])
2
ω
2
(+)
2
a
3
ω
2
Cos[tω]
2
a
3
ω
2
Sin[tω]
Out[]=
bω
2
b
2
a
2
ω
The torsion of the helix is constant.
Animator:
In[]:=
SwatchLegend[{Green,Orange},{"Tangent","Normal"}]ManipulateClearAll;a=2;b=3;ω=2;ζ[t_]:={aCos[ωt],aSin[ωt],bt};θ=NArcCos+;κ=+;χ=+;n[t_]:=-Cos[tω],-Sin[tω],0;p=Grid[{{AngleTangentXY,Curvature,Torsion},{θ,κ,χ}},FrameAll,ItemSize->15,ItemStyle->15];Show[{ParametricPlot3D[ζ[t],{t,0,4π},ColorFunction->"BlueGreenYellow",PlotStyleThickness[0.009]],Graphics3D[{Green,Point[ζ[t1]],Arrowheads[.05],Arrow[Tube[{ζ[t1],ζ[t1]+2ζ'[t1]}]]}],Graphics3D[{Orange,Point[ζ[t1]],Arrowheads[.05],Arrow[Tube[{ζ[t1],ζ[t1]+8n[t1]}]]}],Graphics3D[{Black,Ball[ζ[t1],0.2]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},PlotRangeAll,BoxRatiosAutomatic,AxesTrue,BoxedFalse,PlotRangeAll,PerformanceGoal"Quality",ImageSize500,ViewPoint{-3,3,30}],Style["Helix with its tangents and normals:",Bold,Large],{{t1,0.1,"Value (t1)"},0.01,4π,0.01},Delimiter,{{p,2,"Helix parameters:"}},ControlPlacementTop
180
π
b
2
b
2
a
2
ω
a
2
ω
2
b
2
a
2
ω
bω
2
b
2
a
2
ω
3/2
((+))
2
a
4
ω
2
b
2
a
2
ω
3
a
6
ω
3/2
(+)
2
b
2
a
2
ω
3/2
((+))
2
a
4
ω
2
b
2
a
2
ω
3
a
6
ω
3/2
(+)
2
b
2
a
2
ω
Out[]=
Out[]=
