Wolfram Cloud Document

EnvelopeFamilyCurves::usage="Investigate the curve family:

2

C

2

x

+

2

y

= C xReferences:Fedenko,A.S.ProblemasdeGeometríaDiferencial.Ed.Mir:Moscú,1984.";

Development:

In[]:=

Solve[

2

C

2

x

+

2

y

==Cx,

∂

C

(

2

C

2

x

+

2

y

-Cx==0),{x,y}]

{x0,y0},x

1

2C

,y-

1

2

,x

1

2C

,y

1

2

;

This result is interpreted as follows:The envelope of the family of ellipses are the straight lines: {

1

2C

,-

1

2

,0y



1

2C

,

1

2

,0

and the point {0, 0, 0}.

Animation:

In[]:=

ManipulateShowPlot

1

2

,{x,-10,10},PlotStyle->{Blue,Thick},Plot-

1

2

,{x,-10,10},PlotStyle->{Blue,Thick},Table[ContourPlot[

2

C

2

x

+

2

y

==Cx,{x,-10,10},{y,-1,1},ContourStyle{DarkGreen,Thick},PerformanceGoal"Quality"],{C,-5,k1,0.1}],Graphics[{Black,PointSize[0.01],Point[{0,0}]}],AxesLabel->{x,y},AxesOrigin{0,0},AxesTrue,PlotRange{{-10,10},{-1,1}},ImageSize900,AspectRatioAutomatic,{k1,-5,5,0.1},SaveDefinitionsTrue,ControlPlacementTop

Out[]=

k1

Let’s make the problem more interesting and project orthogonally from the plane z = 0 to the plane x + y + z = 0:

Family of ellipses:

2

x-

1

2C

1

4

2

C

+

2

y

1

4

==1;z==0;

1

2C

Cos[t]+

1

2C

,

1

2

Sin[t],0;

In[]:=

Solvex+y+z==0,

x-

1

2C

Cos[t]+

1

2C

1

==

y-

1

2

Sin[t]

1

==

z

1

,{x,y,z}

x-

-2-2Cos[t]+CSin[t]

6C

,y-

1+Cos[t]-2CSin[t]

6C

,z-

1+Cos[t]+CSin[t]

6C

;ζ[t_]:=-

-2-2Cos[t]+CSin[t]

6C

,-

1+Cos[t]-2CSin[t]

6C

,-

1+Cos[t]+CSin[t]

6C

;

The envelope:

In[]:=

Solvex+y+z==0,

x-

1

2C

1

==

y-

1

2

1

==

z

1

,{x,y,z}

x-

-2+C

6C

,y-

1-2C

6C

,z-

1+C

6C

;

Γ

1

[C_]:=-

-2+C

6C

,-

1-2C

6C

,-

1+C

6C

;

In[]:=

Solvex+y+z==0,

x-

1

2C

1

==

y+

1

2

1

==

z

1

,{x,y,z}

x-

-2-C

6C

,y-

1+2C

6C

,z-

1-C

6C

;

Γ

2

[C_]:=-

-2-C

6C

,-

1+2C

6C

,-

1-C

6C

;

Animation of the projection:

In[]:=

Manipulateζ[t_]:=-

-2-2Cos[t]+CSin[t]

6C

,-

1+Cos[t]-2CSin[t]

6C

,-

1+Cos[t]+CSin[t]

6C

;

ζ

1

[t_]:=

1

2C

Cos[t]+

1

2C

,

1

2

Sin[t],0;

Γ

1

[C_]:=-

-2+C

6C

,-

1-2C

6C

,-

1+C

6C

;

Γ

2

[C_]:=-

-2-C

6C

,-

1+2C

6C

,-

1-C

6C

;σ[x_,y_,z_]:=x+y+z;ShowContourPlot3D[σ[x,y,z]==0,{x,-10,10},{y,-10,10},{z,-10,10},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],ContourStyleOpacity[0.3]],ContourPlot3D[z==0,{x,-10,10},{y,-10,10},{z,-10,10},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],ContourStyleOpacity[0.1]],ParametricPlot3D[

Γ

1

[C],{C,-10,10},PlotStyle{Blue,Thick}],ParametricPlot3D[

Γ

2

[C],{C,-10,10},PlotStyle{Blue,Thick}],Table[ParametricPlot3D[ζ[t],{t,0,2π},PlotStyle{DarkGreen,Thick}],{C,-5,k1,0.3}],Table[ParametricPlot3D[

ζ

1

[t],{t,0,2π},PlotStyle{DarkGreen,Thick}],{C,-5,k1,0.3}],ParametricPlot3D[

Γ

1

[C],{C,-10,10},PlotStyle->{Blue,Thick}],ParametricPlot3D[

Γ

2

[C],{C,-10,10},PlotStyle->{Blue,Thick}],ParametricPlot3D

1

2C

,

1

2

,0,{C,-10,10},PlotStyle->{Blue,Thick},ParametricPlot3D

1

2C

,-

1

2

,0,{C,-10,10},PlotStyle->{Blue,Thick},Graphics3D[{Black,Ball[{0,0,0},0.05]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{40,-10,10.},PlotRange6,ImageSize900,BoxRatiosAutomatic,{k1,-5,5,0.3},SaveDefinitionsTrue,ControlPlacementTop

Out[]=

k1