InterseccionDeSuperficiesTransversalmente::usage="
:References:Mishchenko,A.S.,Solovyev,Yu.P.,Fomenko,A.T.ProblemsinDifferentialGeometryandTopology.MirPublishersMoscow:p.58,1985;Personal.";
Development:
In[]:=
Solve[{
2
x
+
2
y
+
2
z
==a,
2
x
+
2
y
-
2
z
==1},{x,z}]
x-
1+a-2
2
y
2
,z-
-1+a
2
,x-
1+a-2
2
y
2
,z
-1+a
2
,x
1+a-2
2
y
2
,z-
-1+a
2
,x
1+a-2
2
y
2
,z
-1+a
2
;
If a =1, the coordinate z is canceled,therefore a ≠ 1 (a > 1):
In[]:=
Reduce[1+a-2
2
y
>=0,y]
(a-1&&y0)||a>-1&&-
1+a
2
≤y≤
1+a
2
;
In[]:=
​​​​Manipulate​​ShowContourPlot3D[
2
x
+
2
y
-
2
z
==1,{x,-5,5},{y,-5,5},{z,-4,4},ContourStyleDirective[Yellow,Opacity[0.9],Specularity[White,30]],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality"],​​ParametricPlot3D[
a
Cos[u]Sin[v],
a
Sin[u]Sin[v],
a
Cos[v],{u,0,2π},{v,0,π},PlotStyleDirective[Red,Opacity[0.7],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],​​ParametricPlot3D-
1+a-2
2
y
2
,y,-
-1+a
2
,y,-
1+a
2
,
1+a
2
,PlotStyle{Thickness[0.005],Blue},ParametricPlot3D
1+a-2
2
y
2
,y,-
-1+a
2
,y,-
1+a
2
,
1+a
2
,PlotStyle{Thickness[0.005],Blue},ParametricPlot3D-
1+a-2
2
y
2
,y,
-1+a
2
,y,-
1+a
2
,
1+a
2
,PlotStyle{Thickness[0.005],Blue},ParametricPlot3D
1+a-2
2
y
2
,y,
-1+a
2
,y,-
1+a
2
,
1+a
2
,PlotStyle{Thickness[0.005],Blue}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,1},ImageSize900,Style["Surfaces that intersect transversely",Bold,Large],{{a,1.00002,"Value (a)"},1.00001,15,0.00001},ControlPlacementTop,SaveDefinitions->True​​
Out[]=
​
Surfaces that intersect transversely
Value (a)