ImageCurveSurface::usage="Demonstrate that the image of the curve x =
t​

Cos[t]​, y =
t

Sin[t]​, z = 2t is on the surface
2
x
+
2
y
-
z

= 0 and the osculating plane of the curve matches the tangent plane of the surface. References:Fedenko,A.S.ProblemasdeGeometríaDiferencial.EditorialMir:p.64,Moscú,1981.;Personal.";
Development:
Demonstration that the curve is on the surface:
In[]:=
2
(
t

Cos[t])
+
2
(
t

Sin[t])
-
2t

==0
-
2t

+
2t

2
Cos[t]
+
2t

2
Sin[t]
0;-
2t

+
2t

(
2
Cos[t]
+
2
Sin[t]
)==-
2t

+
2t

==0
Demonstration that the osculating plane coincides with the tangent plane. Be the point t = 0, P = {1, 0, 0}:
In[]:=
r[t_]:={
t

Cos[t],
t

Sin[t],2t};​​ResourceFunction["TangentPlane"][
2
x
+
2
y
-
z

==0,{x,1},{y,0},z]
Out[]=
StandardFormEquation02(-1+x)-z,Slopesx2,y0,z-1
2x-z-2==0;
In[]:=
ResourceFunction["OsculatingPlane"][{
t

Cos[t],
t

Sin[t],2t},t]
InfinitePlane{
t

Cos[t],
t

Sin[t],2t},
t

Cos[t]-
t

Sin[t]
2
2+
2t

,
t

Cos[t]+
t

Sin[t]
2
2+
2t

,
2
2+
2t

,-
t

2t

(4+
2t

)
3
Cos[t]
2
2+
2t

(4+
2t

)
-
2
2
-t

2t

(4+
2t

)
Sin[t]
2+
2t

(4+
2t

)
-
t

2t

(4+
2t

)
2
Cos[t]
Sin[t]
2
2+
2t

(4+
2t

)
-
t

2t

(4+
2t

)
Cos[t]
2
Sin[t]
2
2+
2t

(4+
2t

)
-
t

2t

(4+
2t

)
3
Sin[t]
2
2+
2t

(4+
2t

)
,
2
2
-t

2t

(4+
2t

)
Cos[t]
2+
2t

(4+
2t

)
+
t

2t

(4+
2t

)
3
Cos[t]
2
2+
2t

(4+
2t

)
-
t

2t

(4+
2t

)
2
Cos[t]
Sin[t]
2
2+
2t

(4+
2t

)
+
t

2t

(4+
2t

)
Cos[t]
2
Sin[t]
2
2+
2t

(4+
2t

)
-
t

2t

(4+
2t

)
3
Sin[t]
2
2+
2t

(4+
2t

)
,-
2
2t

(4+
2t

)
2
Cos[t]
2+
2t

(4+
2t

)
-
2
2t

(4+
2t

)
2
Sin[t]
2+
2t

(4+
2t

)
/.t0
InfinitePlane{1,0,0},
1
6
,
1
6
,
2
3
,-
1
30
,2
2
15
+
1
30
,-
2
15
;
In[]:=
Eliminatex==1+
1
6
u-
1
30
v,y==
1
6
u+2
2
15
+
1
30
v,z==
2
3
u-
2
15
v,{u,v}
2
6
+
6
z2
6
x;
2x-z-2==0;
In[]:=
​​Manipulate[r[t_]:={
t

Cos[t],
t

Sin[t],2t};p=ResourceFunction["OsculatingPlane"][r[t],t];​​Show[{Graphics3D[{Yellow,Opacity[0.3],p}]/.tt1,​​ContourPlot3D[
2
x
+
2
y
-
z

==0,{x,-4,4},{y,-4,4},{z,-4,4},Mesh->None,ContourStyleDirective[Green,Opacity[0.1],Specularity[White,30]],PerformanceGoal"Quality"],ParametricPlot3D[r[t],{t,-2,t1},PlotStyle{Thick,Blue}],​​Graphics3D[{Black,Ball[r[t1],0.1]}]}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,-1},ImageSize500,PerformanceGoal"Quality"],Style["Curve and osculating and tangent planes coincide at point P:",Bold,Large],{{t1,-1.9,"Value (t1)"},-2+0.00001,2,0.00001},ControlPlacementTop]
Out[]=
​
Curve and osculating and tangent planes coincide at point P:
Value (t1)