TranslationSurface::usage=" The translation surface of the following two parabolas will be shown: = 2 p x; z = 0; and = 2 q x; y = 0.References: https://fernandorevilla.es; Personal";
2
y
2
z
Cartesian equation of translation surface:
ζ[u_]:=,u,0;[v_]:=,0,v;
2
u
2p
*
ζ
2
v
2q
They intersect at the point {0, 0, 0} because both parabolas have their vertex at the origin.
In[]:=
Eliminatex==+,y==u,z==v,{u,v}
2
u
2p
2
v
2q
-2pqx+q+p0;
2
y
2
z
For the animation we will take p = q = 2.
Manipulatep=2;q=2;ζ[u_]:=,u,0;[v_]:=,0,v;σ[u_,v_]:=+,u,v;Show[ParametricPlot3D[σ[u,v],{u,-5,5},{v,-5,k1},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],PlotStyleOpacity[0.1]],ParametricPlot3D[ζ[u],{u,-5,5},PlotStyle->Black],ParametricPlot3D[[v],{v,-5,5},PlotStyle->Black],Graphics3D[{Black,Ball[{0,0,0},0.18]}],Graphics3D[{Blue,Ball[[k1],0.18]}],ParametricPlot3D[σ[u,k1],{u,-5,5}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,PlotRangeAll,ViewPoint{1.3,-4,-1.}],{k1,-5+0.00001,5,0.00001},SaveDefinitionsTrue,ControlPlacementTop
2
u
2p
*
ζ
2
v
2q
2
u
2p
2
v
2q
*
ζ
*
ζ
Out[]=
