AnalysisOfAHyperbolicParaboloidWithDifferentialGeometry::usage="Surface analysis z = x y in the point P = {0, 0, 0} with Differential Geometry.Reference: Personal.";
Development:
In[]:=
s[u_,v_]:={u,v,uv};​​ResourceFunction["PrincipalCurvatures"][s[u,v],{u,v}]​​
Out[]=

(1+
2
u
)(1+
2
v
)
3
(1+
2
u
+
2
v
)
-
uv
3/2
(1+
2
u
+
2
v
)
,-
(1+
2
u
)(1+
2
v
)
3
(1+
2
u
+
2
v
)
-
uv
3/2
(1+
2
u
+
2
v
)

Principal curvatures:
In[]:=

(1+
2
u
)(1+
2
v
)
3
(1+
2
u
+
2
v
)
-
uv
3/2
(1+
2
u
+
2
v
)
,-
(1+
2
u
)(1+
2
v
)
3
(1+
2
u
+
2
v
)
-
uv
3/2
(1+
2
u
+
2
v
)
/.{u0,v0}
Out[]=
{1,-1}
The P point is hyperbolic, because
K
G
<0.
First fundamental form:
In[]:=
ResourceFunction["FirstFundamentalFormCoefficients"][s[u,v],{u,v}]
Out[]=
{1+
2
v
,uv,1+
2
u
}
In[]:=
{1+
2
v
,uv,1+
2
u
}/.{u0,v0}
Out[]=
{1,0,1}
Second form fundamental:
In[]:=
ResourceFunction["SecondFundamentalFormCoefficients"][s[u,v],{u,v}]
Out[]=
0,
1
-
2
u
2
v
+(1+
2
u
)(1+
2
v
)
,0
In[]:=
0,
1
-
2
u
2
v
+(1+
2
u
)(1+
2
v
)
,0/.{u0,v0}
Out[]=
{0,1,0}
Obtaining the princial directions:
In[]:=
EE=1;F=0;G=1;l1=0;f1=1;g1=0;​​Solve[Det[{{
2
λ
,-λ,1},{EE,F,G},{l1,f1,g1}}]==0,λ]​​
Out[]=
{{λ-1},{λ1}}
The directions are indicated as (1, λ) and the vectors will appear green colored in the graph:
In[]:=
s
u
[u_,v_]=D[s[u,v],u];​​
s
v
[u_,v_]=D[s[u,v],v];​​V=1
s
u
[0,0]-1
s
v
[0,0]​​V1=1
s
u
[0,0]+1
s
v
[0,0]
Out[]=
{1,-1,0}
Out[]=
{1,1,0}
The asymptotic directions are indicated as (1, λ) y (λ, 1) and the vectors will appear red colored in the graph:
In[]:=
l1=0;f1=1;g1=0;​​Solve[l1+2f1λ+g1
2
λ
==0,λ]
Out[]=
{{λ0}}
In[]:=
V
A
=1
s
u
[0,0]+0
s
v
[0,0]​​
V
A1
=0
s
u
[0,0]+1
s
v
[0,0]
Out[]=
{1,0,0}
Out[]=
{0,1,0}
Normal to the surface (blue colored) and the tangent plane (blue colored):
In[]:=
ρ[u_,v_]=s[u,v]+α
s
u
[u,v]+β
s
v
[u,v];​​ρ[0,0]​​N1=
s
u
[0,0]
s
v
[0,0]
Norm[
s
u
[0,0]
s
v
[0,0]]
Out[]=
{α,β,0}
Out[]=
{0,0,1}
In[]:=
​​Manipulate[ClearAll;s[u_,v_]:={u,v,uv};​​Show[{ParametricPlot3D[s[u,v],{u,-2,t1},{v,-2,t1},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality"],ParametricPlot3D[s[0,v],{v,-2,t1},PlotStyleBlack],​​ParametricPlot3D[s[u,0],{u,-2,t1},PlotStyleBlack],Graphics3D[{Red,Ball[{0,0,0},0.08]}],Graphics3D[{Green,Point[{0,0,0}],Arrowheads[.05],Arrow[Tube[{{0,0,0},{0,0,0}+(t1+2){1,-1,0}}]]}],​​Graphics3D[{Green,Point[{0,0,0}],Arrowheads[.05],Arrow[Tube[{{0,0,0},{0,0,0}+(t1+2){1,1,0}}]]}],​​Graphics3D[{Red,Point[{0,0,0}],Arrowheads[.05],Arrow[Tube[{{0,0,0},{0,0,0}+(t1+2){1,0,0}}]]}],​​Graphics3D[{Red,Point[{0,0,0}],Arrowheads[.05],Arrow[Tube[{{0,0,0},{0,0,0}+(t1+2){0,1,0}}]]}],Graphics3D[{Blue,Point[{0,0,0}],Arrowheads[.05],Arrow[Tube[{{0,0,0},{0,0,0}+(t1+2){0,0,1}}]]}],ParametricPlot3D[{α,β,0},{α,-2,t1},{β,-2,t1},PlotStyle->Opacity[0.1],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow"]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-3,5,3}],Style["Ruled surface:",Bold,Medium],{{t1,1,"Value (t1)"},-2+0.01,2,0.01},ControlPlacementTop]
Out[]=
​
Ruled surface:
Value (t1)
The surface for being a hyperbolic paraboloid is alabed.