StrictionLine::usage="Given the surface x =
2
u
+ u v, y = u +
2
u
v, z = u + v, find the equation of the striction line. References: Usunariz B., Ubaldo. Problemas de Geometría analítica y ​​ Diferencial;Personal ";
Development:
In[]:=
s[u_,v_]:={
2
u
+uv,u+
2
u
v,u+v};c[u_]:={
2
u
,u,u};w[u_]:={u,
2
u
,1};
Demonstration that it is not a developable surface:
In[]:=
ResourceFunction["PrincipalCurvatures"][s[u,v],{u,v}]
In[]:=

-
6
u
+2
3
u
v+7
5
u
v-
2
v
-
2
u
2
v
-
4
u
(-9+
2
v
)
3/2
(1+4
6
u
+6uv-2
3
u
v-4
5
u
v+
2
v
+4
2
u
2
v
+
4
u
(-3+
2
v
))
+((1+
2
u
+
4
u
)(1+
8
u
+6uv-14
7
u
v+
2
v
+
4
v
-4
3
u
v(11+
2
v
)+17
6
u
(1+3
2
v
)-2
5
u
v(-50+7
2
v
)+
4
u
(12-30
2
v
+
4
v
)+
2
u
(-7-3
2
v
+
4
v
)))
3
(1+4
6
u
+6uv-2
3
u
v-4
5
u
v+
2
v
+4
2
u
2
v
+
4
u
(-3+
2
v
))
,
-
6
u
+2
3
u
v+7
5
u
v-
2
v
-
2
u
2
v
-
4
u
(-9+
2
v
)
3/2
(1+4
6
u
+6uv-2
3
u
v-4
5
u
v+
2
v
+4
2
u
2
v
+
4
u
(-3+
2
v
))
-((1+
2
u
+
4
u
)(1+
8
u
+6uv-14
7
u
v+
2
v
+
4
v
-4
3
u
v(11+
2
v
)+17
6
u
(1+3
2
v
)-2
5
u
v(-50+7
2
v
)+
4
u
(12-30
2
v
+
4
v
)+
2
u
(-7-3
2
v
+
4
v
)))
3
(1+4
6
u
+6uv-2
3
u
v-4
5
u
v+
2
v
+4
2
u
2
v
+
4
u
(-3+
2
v
))
/.{u1,v1}
In[]:=
N
7
8
2
+
57
2
8
,
7
8
2
-
57
2
8

Out[]=
{1.28604,-0.048599}
The
K
G
≠0,p[u]
≠ 0, it is not a developable surface, the point P = {u = 1, v = 1} or P = {2, 2, 2} it’s a hyperbolic point.
Striction line:
s[u,v]=c[u]+vw[u];
In[]:=
β[u]=c[u]-
(c'[u]w[u]).(w'[u]w[u])
2
(Norm[w'[u]w[u]])
w[u]
β[u_]:=
2
u
-
u(u+2u(1-
2
u
)-
2
u
(-u+2
3
u
))
1+4
2
u
+
4
u
,u-
2
u
(u+2u(1-
2
u
)-
2
u
(-u+2
3
u
))
1+4
2
u
+
4
u
,u-
u+2u(1-
2
u
)-
2
u
(-u+2
3
u
)
1+4
2
u
+
4
u
;
In[]:=
​​ManipulateClearAll;s[u_,v_]:={
2
u
+uv,u+
2
u
v,u+v};β[u_]:=
2
u
-
u(u+2u(1-
2
u
)-
2
u
(-u+2
3
u
))
1+4
2
u
+
4
u
,u-
2
u
(u+2u(1-
2
u
)-
2
u
(-u+2
3
u
))
1+4
2
u
+
4
u
,u-
u+2u(1-
2
u
)-
2
u
(-u+2
3
u
)
1+4
2
u
+
4
u
;​​Show[{ParametricPlot3D[s[u,v],{u,-3,3t1},{v,-3,3t1},PlotStyle->Opacity[0.3],ColorFunction"BlueGreenYellow",MeshFalse,PerformanceGoal"Quality",BoundaryStyleDirective[Black,Thick]],​​Graphics3D[{Black,Ball[{2,2,2},0.25]}],ParametricPlot3D[β[u],{u,-2,2t1},PlotStyle{Red,Thick}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,-10,20},ImageSize500],Style["Striction line (red colored):",Bold,Large],{{t1,0,"Value (t1)"},-1+0.01,1,0.01},ControlPlacementTop
Out[]=
​
Striction line (red colored):
Value (t1)