CurveOnSurface::usage="Given the curve {r[θ] Cos[θ]​, r[θ] Sin[θ]​, r[θ]​}​, find r[θ]​, such that its tangent forms a constant angle with the plane XY.References: Usunáriz Balanzategui, Ubaldo and Usunáriz Sala, Ignacio. Problemas de Geometría ​​Analítica y Diferencial.p.333; Personal.";
Development:
In[]:=
ζ[θ_]:={r[θ]Cos[θ],r[θ]Sin[θ],r[θ]};​​ζ'[θ]​​
{-r[θ]Sin[θ]+Cos[θ]
′
r
[θ],Cos[θ]r[θ]+Sin[θ]
′
r
[θ],
′
r
[θ]};
Tangent line to the curve:
ℒ:ζ[θ]+t{-r[θ]Sin[θ]+Cos[θ]
′
r
[θ],Cos[θ]r[θ]+Sin[θ]
′
r
[θ],
′
r
[θ]};
Angle with XY plane:
In[]:=
Cos[α]==
{0,0,1}.{-r[θ]Sin[θ]+Cos[θ]
′
r
[θ],Cos[θ]r[θ]+Sin[θ]
′
r
[θ],
′
r
[θ]}
Norm[{-r[θ]Sin[θ]+Cos[θ]
′
r
[θ],Cos[θ]r[θ]+Sin[θ]
′
r
[θ],
′
r
[θ]}].Norm[{0,0,1}]
In[]:=
SimplifyCos[α]
′
r
[θ]
2
(
′
r
[θ])
+
2
(-r[θ]Sin[θ]+Cos[θ]
′
r
[θ])
+
2
(Cos[θ]r[θ]+Sin[θ]
′
r
[θ])
.1

Cos[α]
′
r
[θ]
2
r[θ]
+2
2
′
r
[θ]
==c;
Where c it is a constant.
In[]:=
Solve
′
r
[θ]
2
r[θ]
+2
2
′
r
[θ]
==c,r[θ]
r[θ]-
2
′
r
[θ]
-2
2
c
2
′
r
[θ]
c
,r[θ]
2
′
r
[θ]
-2
2
c
2
′
r
[θ]
c
;
r[θ]=±r'[θ]
1-2
2
c
c
=±r'[θ]
c
1
;
Where
c
1
it is a constant.
In[]:=
DSolve[r[θ]==r'[θ]
c
1
,r[θ],θ]
r[θ]
θ
c
1


1
;
The curve sought is:
ζ[θ]=±
θ
c
1


1
Cos[θ],±
θ
c
1


1
Sin[θ],±
θ
c
1


1
;
For the graph
c
1
= 1 and consider positive sign:
​​ManipulateClearAll;ζ[θ_]:=
θ

Cos[θ],
θ

Sin[θ],
θ

;r[θ_]:=
θ

;α[t1_]:=N
180
π
ArcCos
′
r
[t1]
2
r[t1]
+2
2
′
r
[t1]
;​​p=Grid[{{AngleTangentWithXY},{α[t1]}},FrameAll,ItemSize->15,ItemStyle->15];​​ShowContourPlot3D[z==0,{x,-6,6},{y,-6,6},{z,-2,2},ContourStyle->{Opacity[0.2],Yellow},MeshFalse,PerformanceGoal"Quality"],ParametricPlot3Dζ[t1]+tζ'[t1],t,-2,
t1
π
,PlotStyle{Thick,Green},​​ParametricPlot3D[ζ[θ],{θ,0,t1},PlotStyle{Blue,Thick}],Graphics3D[{Green,Ball[ζ[t1],0.15]}],Graphics3D[{Orange,Ball[{
t1

Cos[t1],
t1

Sin[t1],0},0.15]}],Graphics3D[{Green,Point[ζ[t1]],Arrowheads[.05],Arrow[Tube[{ζ[t1],ζ[t1]+0.5ζ'[t1]}]]}],Graphics3D[{Orange,Point[{
t1

Cos[t1],
t1

Sin[t1],0}],Arrowheads[.05],Arrow[Tube[{{
t1

Cos[t1],
t1

Sin[t1],0},{
t1

Cos[t1],
t1

Sin[t1],0}+8{0,0,1}}]]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,2},ImageSize700,Style["Curve on a surface forms a constant angle with the XY plane:",Bold,Large],{t1,0.1,"Value (t1)"},0.01,
π
2
,0.01,Delimiter,{{p,2,"Angle that form the line tangent to the curve and the plane z = 0"}},ControlPlacementTop,SaveDefinitions->True
Out[]=
​
Curve on a surface forms a constant angle with the XY plane:
Value (t1)
Angle that form the line tangent to the curve and the plane z = 0
AngleTangentWithXY
54.7356