SurfaceCircularHelix::usage="Given the circular helix γ[t] = {Cos[t]​, Sin[t]​, t}​, 0 ≤ t < 2π, the tubular surface S is considered, for each value of t, by the points of the circumference
C
t
​, of center γ[t]​, located in the normal plane of γ[t] and radius
2
t
. 1 Parameterize S in the form:
r
​[t, α] = γ[t] +
R
​[t, α]​, being α the central angle, measured over the circumference of S corresponding to the point, from the main normal and in a positive direction around the advance. 2 Determine the first fundamental form of S at the point t = 1, α =
π
4
. References: www2.caminos.upm.es; Personal.";
Development (1):
Normal plane of γ[t]:
In[]:=
γ[t_]:={Cos[t],Sin[t],t};​​(x-Cos[t])(-Sin[t])+(y-Sin[t])Cos[t]+(z-t)(1)==0//FullSimplify​​
t+xSin[t]z+yCos[t];
-t+z+Cos[t](y-Sin[t])-(x-Cos[t])Sin[t]0;​​-Sin[t]x+Cos[t]y+z-t==0;
Circumference
C
t
:
r=
2
t
;​​x=t+Cot[t]y+Csc[t]z-Csc[t]t;​​y=u;z=v;​​{Cot[t],1,0}{-Sin[t],Cos[t],1}​​​​
{1,-Cot[t],Cos[t]Cot[t]+Sin[t]};
C
t
[t_]:=γ[t]+Cos[t]
2
t
Norm[{Cot[t],1,0}]
{Cot[t],1,0}+Sin[t]
2
t
Norm[{1,-Cot[t],Cos[t]Cot[t]+Sin[t]}]
{1,-Cot[t],Cos[t]Cot[t]+Sin[t]};
Main normal of γ[t]::
In[]:=
γ[t_]:={Cos[t],Sin[t],t};​​R=γ[t]+λ((γ'[t]γ''[t])γ'[t])
R[t_]:={Cos[t]+λ(-Cos[t]-
3
Cos[t]
-Cos[t]
2
Sin[t]
),Sin[t]+λ(-Sin[t]-
2
Cos[t]
Sin[t]-
3
Sin[t]
),t};
For t =
π
3
graph it would be this:
In[]:=
Showγ[t_]:={Cos[t],Sin[t],t};
C
t
[t_]:=γ[t]+Cos[t]
2
t
Norm[{Cot[t],1,0}]
{Cot[t],1,0}+Sin[t]
2
t
Norm[{1,-Cot[t],Cos[t]Cot[t]+Sin[t]}]
{1,-Cot[t],Cos[t]Cot[t]+Sin[t]};​​R1[t_]:={Cos[t]+λ(-Cos[t]-
3
Cos[t]
-Cos[t]
2
Sin[t]
),Sin[t]+λ(-Sin[t]-
2
Cos[t]
Sin[t]-
3
Sin[t]
),t};​​ParametricPlot3D
1
2
+
1
18
2
π
Cos[t]+
2
π
Sin[t]
9
4
3
+
2
1
2
3
+
3
2
,
3
2
+
2
π
Cos[t]
6
3
-
2
π
Sin[t]
9
3
4
3
+
2
1
2
3
+
3
2
,
π
3
+
1
2
3
+
3
2
2
π
Sin[t]
9
4
3
+
2
1
2
3
+
3
2
,{t,0,2π},PlotStyle{Yellow,Thick},ContourPlot3D-Sin
π
3
x+Cos
π
3
y+z-
π
3
==0,{x,-2,2},{y,-2,2},{z,-2,2},MeshFalse,ContourStyle{Gray,Opacity[0.2]},ParametricPlot3D[{Cos[t],Sin[t],t},{t,0,2π},PlotStyle->{DarkBlue,Thick}],ParametricPlot3DR1
π
3
,{λ,-2,2},PlotStyle->{Orange,Thick},Graphics3DBlack,Ball
C
t

π
3
,0.03,Graphics3DDarkBlue,Ballγ
π
3
,0.03,Graphics3DBlack,Thick,Dashed,Lineγ
π
3
,
C
t

π
3

Out[]=
Animation to understand the parameterization of the surface:
In[]:=
​​Manipulateγ[t_]:={Cos[t],Sin[t],t};
C
t
[t_]:=γ[t1]+Cos[t]
2
t1
Norm[{Cot[t1],1,0}]
{Cot[t1],1,0}+Sin[t]
2
t1
Norm[{1,-Cot[t1],Cos[t1]Cot[t1]+Sin[t1]}]
{1,-Cot[t1],Cos[t1]Cot[t1]+Sin[t1]};​​R1[t_]:={Cos[t]+λ(-Cos[t]-
3
Cos[t]
-Cos[t]
2
Sin[t]
),Sin[t]+λ(-Sin[t]-
2
Cos[t]
Sin[t]-
3
Sin[t]
),t};​​​​ShowParametricPlot3D[γ[t],{t,0,t1},PlotStyle->DarkBlue]/.Line[pts_,rest___]:>Tube[pts,0.15,rest],​​ParametricPlot3Dγ[t1]+Cos[t]
2
t1
Norm[{Cot[t1],1,0}]
{Cot[t1],1,0}+Sin[t]
2
t1
Norm[{1,-Cot[t1],Cos[t1]Cot[t1]+Sin[t1]}]
{1,-Cot[t1],Cos[t1]Cot[t1]+Sin[t1]},{t,0,2π},PlotStyle->Yellow/.Line[pts_,rest___]:>Tube[pts,0.18,rest],ParametricPlot3D[R1[t1],{λ,-20,20},PlotStyle->Orange]/.Line[pts_,rest___]:>Tube[pts,0.2,rest],ContourPlot3D[-Sin[t1]x+Cos[t1]y+z-t1==0,{x,-40,40},{y,-35,35},{z,-35,35},ContourStyle->{Gray,Opacity[0.2]},MeshFalse],Graphics3D[{Black,Ball[
C
t
[t1],0.45]}],Graphics3D[{DarkBlue,Ball[γ[t1],0.45]}],Graphics3D[{Black,Thick,Dashed,Line[{γ[t1],
C
t
[t1]}]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange40,ViewPoint{1.3,-2.4,4.},ImageSize900,Style["To understand the parameterization:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacementTop
Out[]=
​
To understand the parameterization:
Value (t1)
Parameterization of the surface S in the form of a tube (variable radius). The normal and binormal γ[t] vectors are used:
Development (2):
Finally the animation of the surface S[t, α]: