ConicalHelix::usage="
​Reference:e.math.cornell.edu,DifferentialGeometry,JimBelk,CornellUniversity,Homework5(ParametricCurvesinThreeDimensions),Prob.-2.";
Development:
Equation of the tangent line to the original helix (in black):
ℒ
T
:Cos[t],Sin[t],
1
2
t+v-Sin[t],Cos[t],
1
2
;
Equation of the plane that contains the origin and is perpendicular to the tangent line to the original helix:
In[]:=
{x,y,z}.-Sin[t],Cos[t],
1
2
==0
z
2
+yCos[t]-xSin[t]0;
Coordinates of the intersection point of the plane with the tangent line:
In[]:=
Solve-(Cos[t]-vSin[t])Sin[t]+(Sin[t]+vCos[t])Cos[t]+
1
2
1
2
t+
1
2
v==0,v
v-
t
1+4
2
Cos[t]
+4
2
Sin[t]
;​​v=-
1
5
t;
Parametric equations of the helix we are looking for:
Cos[t]+
1
5
tSin[t],Sin[t]-
1
5
tCos[t],
2
5
t;
​
In[]:=
ManipulateClearAll;c[t_]:=Cos[t],Sin[t],
1
2
t;​​γ[t_]:=Cos[t]+
1
5
tSin[t],Sin[t]-
1
5
tCos[t],
2
5
t;​​α=N
180
π
ArcCos
c'[t1].γ[t1]
Norm[c'[t1]]Norm[γ[t1]]
;​​p=Grid[{{Coordinates,Angle},{γ[t1],α}},FrameAll,ItemSize->20,ItemStyle->15];​​​​Show[{ParametricPlot3D[c[t],{t,0,8π},PlotStyleBlack]/.Line[pts_,rest___]:>Tube[pts,0.09,rest],ParametricPlot3D[γ[t],{t,0,t1},PlotStyleBlue]/.Line[pts_,rest___]:>Tube[pts,0.09,rest],Graphics3D[{Blue,Thick,Ball[γ[t1],0.2]}],Graphics3D[{Red,Thick,Ball[{0,0,0},0.2]}],Graphics3D[{Red,Thick,Line[{{0,0,0},γ[t1]}]}],ParametricPlot3D[c[t1]+vc'[t1],{v,-6,6},PlotStyle{Black,Thick}]​​},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange11,ImageSize900,ViewPoint{-3,3,2}],Style["Conical Helix:",Bold,Medium],{{t1,0.00002,"Value (t1)"},0.00001,8π,0.00001},Delimiter,{{p,2,"Coordinates of the tangent point and angle between straight lines"}},ControlPlacementTop
Out[]=
​
Conical Helix:
Value (t1)
Coordinates of the tangent point and angle between straight lines
Coordinates
Angle
{-4.22498,-2.44104,9.55178}
90.
Original helix in black, helix in blue that results from the rotation of the line around the Z axis.