EquationTangentPlane::usage="Find the equation of the plane tangent to the surface S at point P {1, 0, -1}​, where S is the surface generated by turning the curve C: {z =
2
y
- 2 y, x = 0} around the OZ axis. Obtain the geodesic curvature of C in S. Reason if the intersection curve of S with the plane z = 1 is a geodesic. Reason if there is some asymptotic line of S that passes through the point P. Reference: www2.caminos.upm.es; casanchi.org.mat>cgeodesica01; Personal.";
Development:
How the curve is going to rotate around Z axis, we substitute ±
2
x
+
2
y
in y:
z=
2
x
+
2
y
±2
2
x
+
2
y
;
In[]:=
Expand
2
(z-
2
x
-
2
y
)
==
2
2
2
x
+
2
y


4
x
+2
2
x
2
y
+
4
y
-2
2
x
z-2
2
y
z+
2
z
4
2
x
+4
2
y
;​​
4
x
+2
2
x
2
y
+
4
y
-2
2
x
z-2
2
y
z+
2
z
-4
2
x
-4
2
y
==0;​​
Equation of the plane tangent to the surface at the point P{1, 0, -1}:
In[]:=
F[x_,y_,z_]:=
4
x
+2
2
x
2
y
+
4
y
-2
2
x
z-2
2
y
z+
2
z
-4
2
x
-4
2
y
;​​
F
x
[x_,y_,z_]=D[F[x,y,z],x];​​
F
y
[x_,y_,z_]=D[F[x,y,z],y];​​
F
z
[x_,y_,z_]=D[F[x,y,z],z];​​(x-1)
F
x
[1,0,-1]+y
F
y
[1,0,-1]+(z+1)
F
z
[1,0,-1]==0​​
-4(1+z)0;​​z==-1;
Geodesic curvature of the C in S:​Geodesic curvature (
k
g
measures how much a curve, traced over a surface, deviates from being geodesic (the strightest or shortest possible path on that surface). It can be calculated like this
k
g
=[C'[t],C''[t],N]
, where C’[t] is the tangent vector to C and N the normal vector to surface.
Normal with respect to the surface in P {1, 0, -1}:
In[]:=
N1={
F
x
[1,0,-1],
F
y
[1,0,-1],
F
z
[1,0,-1]}
{0,0,-4};
Tangent vector of the curve C at t = 0 ( C’[t] ):
In[]:=
C1[t_]:={0,t,
2
t
-2t};​​
k
g
=Det[{C1'[0],C1''[0],{0,0,-4}}]
Out[]=
0
At that point t = 0 the C curve is a geodesic.
Intersection curve of S with plane z = 1 is a geodesic?
x,-
3-2
2
-
2
x
,1;x,
3-2
2
-
2
x
,1;x,-
3+2
2
-
2
x
,1;x,
3+2
2
-
2
x
,1;
These two circles are not geodesic, because their geodesic curvature is
k
g
=
1
r
, where r is its radius, that is, the geodesic curvature is not zero.
Below are shown the geodesics of the surface S and it is seen that the curve C is included as we had observed, there is no circumference as geodesic:
In[]:=
geo=ResourceFunction["Geodesic"]u,v,
2
u
+
2
v
+2
2
u
+
2
v
,{u,v},t,{0,0},θ
Out[]=

4u[t]1+
2
u[t]
+
2
v[t]

2
u[t]
2
u[t]
+
2
v[t]
+
2
v[t]
1+
2
u[t]
+
2
v[t]

2
′
u
[t]
2
(
2
u[t]
+
2
v[t]
)
5+4
2
u[t]
+4
2
v[t]
+8
2
u[t]
+
2
v[t]

-
8
2
u[t]
v[t]1+
2
u[t]
+
2
v[t]

′
u
[t]
′
v
[t]
2
(
2
u[t]
+
2
v[t]
)
5+4
2
u[t]
+4
2
v[t]
+8
2
u[t]
+
2
v[t]

+
4u[t]1+
2
u[t]
+
2
v[t]

2
v[t]
2
u[t]
+
2
v[t]
+
2
u[t]
1+
2
u[t]
+
2
v[t]

2
′
v
[t]
2
(
2
u[t]
+
2
v[t]
)
5+4
2
u[t]
+4
2
v[t]
+8
2
u[t]
+
2
v[t]

+
′′
u
[t]0,
4v[t]1+
2
u[t]
+
2
v[t]

2
u[t]
2
u[t]
+
2
v[t]
+
2
v[t]
1+
2
u[t]
+
2
v[t]

2
′
u
[t]
2
(
2
u[t]
+
2
v[t]
)
5+4
2
u[t]
+4
2
v[t]
+8
2
u[t]
+
2
v[t]

-
8u[t]
2
v[t]
1+
2
u[t]
+
2
v[t]

′
u
[t]
′
v
[t]
2
(
2
u[t]
+
2
v[t]
)
5+4
2
u[t]
+4
2
v[t]
+8
2
u[t]
+
2
v[t]

+
4v[t]1+
2
u[t]
+
2
v[t]

2
v[t]
2
u[t]
+
2
v[t]
+
2
u[t]
1+
2
u[t]
+
2
v[t]

2
′
v
[t]
2
(
2
u[t]
+
2
v[t]
)
5+4
2
u[t]
+4
2
v[t]
+8
2
u[t]
+
2
v[t]

+
′′
v
[t]0,u[0]0,v[0]0,
′
u
[0]Cos[θ],
′
v
[0]Sin[θ]
In[]:=
solg=Flatten(NDSolve[#1,{u,v},{t,1,4}]&)/@Tablegeo,θ,
π
3
,2π,0.1,1;​​ParametricPlot3DEvaluateu[t],v[t],
2
u[t]
+
2
v[t]
+2
2
u[t]
+
2
v[t]
/.solg,{t,1,4}
Out[]=
Pass some asymptotic line through the point P?
The point P{1, 0, -1} is a parabolic point, therefore if there must be only one asymptotic line that passes through there.
In[]:=
Solve[
4
x
+2
2
x
2
y
+
4
y
-2
2
x
z-2
2
y
z+
2
z
-4
2
x
-4
2
y
==0,z]
Out[]=
z
2
x
+
2
y
-2
2
x
+
2
y
,z
2
x
+
2
y
+2
2
x
+
2
y

In[]:=
σ[u_,v_]:=u,v,
2
u
+
2
v
-2
2
u
+
2
v
;​​ResourceFunction["SecondFundamentalFormCoefficients"][σ[u,v],{u,v}]/.{u1,v0}​​
Out[]=
{2,0,0}
In[]:=
DSolve[2
2
u'[v]
==0,u[v],v]
Out[]=
{{u[v]

1
}}
As expected from a parabolic point only a real asymptotic like passes.
Verification that P{1, 0, -1} is a parabolic point:
LN-
2
M
==0;​​(2)(0)-0==0;
Animation:
In[]:=
ManipulateShowRevolutionPlot3D[{0,y,
2
y
-2y},{y,-5,5},{θ,0,θ1},RevolutionAxis{0,0,1},PlotStyle->Opacity[0.2],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow"],ParametricPlot3D[t{0,0,1},{t,-5,10},ColorFunction->"BlueGreenYellow"],ContourPlot3D[z==-1,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->{Yellow,Opacity[0.6]}],Graphics3D[{Black,Ball[{1,0,-1},0.15]}],ParametricPlot3Dx,-
3-2
2
-
2
x
,1,x,-
3-2
2
,
3-2
2
,PlotStyle{Orange,Thick},ParametricPlot3Dx,
3-2
2
-
2
x
,1,x,-
3-2
2
,
3-2
2
,PlotStyle{Orange,Thick},ParametricPlot3Dx,-
3+2
2
-
2
x
,1,x,-
3+2
2
,
3+2
2
,PlotStyle{Orange,Thick},ParametricPlot3Dx,
3+2
2
-
2
x
,1,x,-
3+2
2
,
3+2
2
,PlotStyle{Orange,Thick},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange8,ViewPoint{1.3,-1,1},ImageSize900,Style["Surface of revolution:",Bold,Medium],{{θ1,0.00002,"Value (θ)"},0.00001,2π,0.00001},ControlPlacementLeft
Out[]=
​
Surface of revolution:
Value (θ)