CurveIntersectionSurfaceAndPlane::usage="Let S: ρ = 1 +
2
z
ρ =
2
x
+
2
y
​ be a surface in
3

and let γ the curve intersection of S with the plane z = y - 1. Calculate a tangent vector to γ at point Q of cartesian coordinates {0, 1, 0} and the normal curvature of γ in Q. References: www2.caminos.upm.es; Personal. ";
Development:
Obtaining the intersection curve:
In[]:=
f[x_,y_]:=
2
x
+
2
y
-1
;​​j[x_,y_]:=y-1;​​f[rCos[t],rSin[t]]==j[rCos[t],rSin[t]];​​roft=r/.First@Solve[%,r]//Simplify;​​z[t_]:=j[roftCos[t],roftSin[t]];​​γ[t_]:={roftCos[t],roftSin[t],z[t]};​​Print[γ[t]]

1
2
Cot[t]Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t],
1
2
Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t],-1+
1
2
Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t]
In[]:=
γ[t_]:=
1
2
Cot[t]Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t],
1
2
Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t],-1+
1
2
Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t];
Tangent vector:
In[]:=
Reduce-1+
1
2
Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t]==0,t
Out[]=

1
∈&&t-
π
2
+2π

1
In[]:=
ResourceFunction["TangentVector"][γ[t],t]/.t-
π
2
Out[]=
{-1,0,0}
Normal curvature:
k
n
=k(
n
.N)
In[]:=
ResourceFunction["Curvature"][γ[t],t]/.t-
π
2
2
;
k=
2
;
In[]:=
ResourceFunction["NormalVector"][γ[t],t]/.t-
π
2
0,
153
2
-77
2
,
153
2
-77
2
;​​
n
=0,
153
2
-77
2
,
153
2
-77
2
;
In[]:=
F[x_,y_,z_]:=
2
x
+
2
y
-1-
2
z
;​​
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];​​N1={
F
x
[0,1,0],
F
y
[0,1,0],
F
z
[0,1,0]}
Out[]=
{0,1,0}
In[]:=
k
n
=
2
0,
153
2
-77
2
,
153
2
-77
2
.{0,1,0}//FullSimplify
-1;​​
K
n
=-1;
Although it is not requested, here we add the normal plane (subtended by the tangent vector to the curve and the normal vector to the surface):
Ψ:{0,1,0}+s{-1,0,0}+w{0,1,0};
Graph:
In[]:=
SwatchLegend[{Blue,Orange,Gray,LightGreen,LightRed},{"Tangent vector","Normal vector to surface","Normal vector to curve","Intersection plane","Normal plane"}]​​γ[t_]:=
1
2
Cot[t]Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t],
1
2
Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t],-1+
1
2
Csc[t]-1+
-1+2Cos[2t]-4Sin[t]
+2Sin[t];​​g=ShowParametricPlot3D[γ[t],{t,0,2π},PlotStyle->{Black,Thick}]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],Graphics3D[{Black,Thick,Ball[{0,1,0},0.06]}],​​Graphics3D[{Blue,Point[{0,1,0}],Arrowheads[.03],Arrow[Tube[{{0,1,0},{0,1,0}+2{-1,0,0}}]]}],​​Graphics3D[{Orange,Point[{0,1,0}],Arrowheads[.03],Arrow[Tube[{{0,1,0},{0,1,0}+2{0,1,0}}]]}],​​Graphics3DGray,Point[{0,1,0}],Arrowheads[.03],ArrowTube{0,1,0},{0,1,0}+20,
153
2
-77
2
,
153
2
-77
2
,ContourPlot3D[z==y-1,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyle->{Green,Opacity[0.1]},PerformanceGoal->"Quality",MeshFalse],​​​​ParametricPlot3D[{0,1,0}+s{-1,0,0}+w{0,1,0},{s,-2,2},{w,-2,2},PlotStyle->{Red,Opacity[0.1]},PerformanceGoal->"Quality",MeshFalse],ContourPlot3D
2
x
+
2
y
==1+
2
z
,{x,-2,2},{y,-2,2},{z,-2,2},ContourStyle->{Yellow,Opacity[0.3]},PerformanceGoal->"Quality",MeshFalse,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-2,2},{-2,3},{-2,2}},ViewPoint{-10,50,15},ImageSize900
Out[]=
Tangent vector
Normal vector to surface
Normal vector to curve
Intersection plane
Normal plane
Out[]=