AsymptoticPlane::usage="Given the surface {u v +
2
u
​,
2​
u
v + u, u + v}​, determine the equation of the asymptotic plane of any generatrix, and in particular for u = 1. References: Usunáriz Balanzategui, Ubaldo. Usunáriz Sala, Ignacio. Problemas ​​ de Geometría Analítica y Diferencial. p. 341; Personal. ";
Development:
The tangent plane of a ruled surface along a generatrix has two identical limit positions at both ends of the generatrix, those are the asymptotic planes related to the generatrix.
In[]:=
σ[u_,v_]:={
2
u
,u,u}+v{u,
2
u
,1};​​Det[{{x-
2
u
,y-u,z-u},{u,
2
u
,1},{1,2u,0}}]==0
Out[]=
-u+
3
u
-2ux+y+
2
u
z0
Simplification:
In[]:=
u=1;​​-u+
3
u
-2ux+y+
2
u
z0
-2x+y+z0;
In[]:=
​​Manipulate[ClearAll;σ[u_,v_]:={
2
u
,u,u}+v{u,
2
u
,1};​​Show[{ContourPlot3D[-2x+y+z0,{x,-6,6},{y,-6,6},{z,-6,6t1},ContourStyle->{Opacity[0.6],Yellow},MeshFalse,PerformanceGoal"Quality"],​​ParametricPlot3D[σ[u,v],{u,-6,6},{v,-6,6},PlotStyle->{Opacity[0.2],Red},MeshFalse,PerformanceGoal"Quality"],ParametricPlot3D[σ[1,v],{v,-6,6},PlotStyle->{Purple,Thick}]​​},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange3,ViewPoint{6,-2.4,20.},ImageSize500],Style["Asymptotic plane for u = 1:",Bold,Large],{{t1,0.1,"Value (t1)"},0.01,1,0.01},ControlPlacementTop]
Out[]=
​
Asymptotic plane for u = 1:
Value (t1)