PedalSurfaceofaLevelSurface::usage="Given the function f[x, y, z] = x y z, obtain for the level surface x y z = 1 its corresponding pedal surface with respect to the origin.Rederences: Problemas de Geometría Analítica y Diferencial, Ubaldo Usunáriz Balanzategui, Ignacio Usunáriz Sala, Prob.- H5; Personal";
Development:
Equation of the tangent plane at the point P (α, β, γ):
f[x_,y_,z_]:=xyz-1;​​
f
x
[x_,y_,z_]=D[f[x,y,z],x];​​
f
y
[x_,y_,z_]=D[f[x,y,z],y];​​
f
z
[x_,y_,z_]=D[f[x,y,z],z];​​(x-α)
f
x
[x,y,z]+(y-β)
f
y
[x,y,z]+(z-γ)
f
z
[x,y,z]==0
yz(x-α)+xz(y-β)+xy(z-γ)0;​​1-yzα+1-xzβ+1-xyγ==0;​​βγx+αγy+αβz-3==0;
Equation of the normal line:
x
βγ
==
y
αγ
==
z
αβ
;
Parametric equations of the Pedal Surface:
Solve
x
βγ
==
y
αγ
==
z
αβ
,βγx+αγy+αβz-3==0,{x,y,z}
x
3βγ
2
α
2
β
+
2
α
2
γ
+
2
β
2
γ
,y
3αγ
2
α
2
β
+
2
α
2
γ
+
2
β
2
γ
,z
3αβ
2
α
2
β
+
2
α
2
γ
+
2
β
2
γ
;​​
Cartesian equation of the Pedal Surface:
Solvex==
3βγ
2
α
2
β
+
2
α
2
γ
+
2
β
2
γ
,y==
3αγ
2
α
2
β
+
2
α
2
γ
+
2
β
2
γ
,z==
3αβ
2
α
2
β
+
2
α
2
γ
+
2
β
2
γ
,{α,β,γ}
α
3
y
z
3
x
+x
2
y
+x
2
z
,β
3
x
z
y
x(
2
x
+
2
y
+
2
z
)
,γ
3
x
y
z
x(
2
x
+
2
y
+
2
z
)
,α-
3
y
z
3
x
+x
2
y
+x
2
z
,β-
3
x
z
y
x(
2
x
+
2
y
+
2
z
)
,γ-
3
x
y
z
x(
2
x
+
2
y
+
2
z
)
;
It is fulfilled that α β γ == 1,
2
α
2
β
2
γ
==1
:
3yz
3
x
+x
2
y
+x
2
z
3xz
2
x
y+
3
y
+y
2
z
3xy
2
x
z+z
2
y
+
3
z
==1;​​
27
2
x
2
y
2
z
xyz
3
(
2
x
+
2
y
+
2
z
)
==1;​​
3
(
2
x
+
2
y
+
2
z
)
-27xyz==0;​​
In[]:=
​​Manipulatef[x_,y_,z_]:=xyz-1;
f
p
[x_,y_,z_]:=
3
(
2
x
+
2
y
+
2
z
)
-27xyz;​​Show[ContourPlot3D[f[x,y,z]==0,{x,-2,2},{y,-2,2},{z,-2,2},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],ContourStyleOpacity[0.1]],​​ContourPlot3D[
f
p
[x,y,z]==0,{x,-2,k1},{y,-2,2},{z,-2,2},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],ContourStyleOpacity[0.6]],Graphics3D[{Black,Ball[{0,0,0},0.06]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,1.},PlotRangeAll,ImageSize500,BoxRatiosAutomatic​​],{k1,-2+0.00001,2,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
​
k1
We make a cut with the plane x = y on the original surface and we get a curve, then we get its pedal curve:
In[]:=
ζ[t_]:=t,t,
1
2
t
;​​ζ'[t]
1,1,-
2
3
t
;
Equation of the orthogonal plane to the tangent line (goes through the origin):
x+y-
2
3
t
z==0;​​
3
t
x+
3
t
y-2z==0;
Parametric equations of the Pedal Curve:
In[]:=
Solve
3
t
x+
3
t
y-2z==0,
x-t
3
t
==
y-t
3
t
==
z-
1
2
t
-2
,{x,y,z}//FullSimplify
x
3t
2+
6
t
,y
3t
2+
6
t
,z
3
4
t
2+
6
t
;
ζ
11
[t_]:=
3t
2+
6
t
,
3t
2+
6
t
,
3
4
t
2+
6
t
;
​
We make a cut with the plane x = - y on the original surface and we get a curve, then we get its pedal curve:
In[]:=
ζ
1
[t_]:=-t,t,-
1
2
t
;​​
ζ
1
'[t]
-1,1,
2
3
t
;
3
t
x-
3
t
y-2z==0;
Parametric equations of the Pedal Curve:
In[]:=
Solve
3
t
x-
3
t
y-2z==0,
x+t
3
t
==
y-t
-
3
t
==
z+
1
2
t
-2
,{x,y,z}//FullSimplify
x-
3t
2+
6
t
,y
3t
2+
6
t
,z-
3
4
t
2+
6
t
;
ζ
22
[t_]:=-
3t
2+
6
t
,
3t
2+
6
t
,-
3
4
t
2+
6
t
;
In[]:=
​​SwatchLegend[{Black,Red,Blue,DarkGreen},{"Original curves","Pedal curve to the original curve","Tangent line to each original curve","Orthogonal line to each tangent line"}]​​Manipulateζ[t_]:=t,t,
1
2
t
;
ζ
1
[t_]:=-t,t,-
1
2
t
;
ζ
11
[t_]:=
3t
2+
6
t
,
3t
2+
6
t
,
3
4
t
2+
6
t
;
ζ
22
[t_]:=-
3t
2+
6
t
,
3t
2+
6
t
,-
3
4
t
2+
6
t
;​​ℒ[t_]:=ζ[t]+vζ'[t];​​
ℒ
1
[t_]:=
ζ
1
[t]+v
ζ
1
'[t];​​​​Show[ParametricPlot3D[ζ[t],{t,-π,π},PlotStyle{Black,Thick}],ParametricPlot3D[
ζ
1
[t],{t,-π,π},PlotStyle{Black,Thick}],​​Graphics3D[{Black,Ball[ζ[k1],0.07]}],Graphics3D[{Black,Ball[
ζ
1
[k1],0.07]}],Graphics3D[{Black,Ball[{0,0,0},0.07]}],​​ParametricPlot3D[
ζ
11
[t],{t,-π,π},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],ParametricPlot3D[
ζ
22
[t],{t,-π,π},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],​​Graphics3D[{Red,Ball[
ζ
11
[k1],0.07]}],Graphics3D[{Red,Ball[
ζ
22
[k1],0.07]}],​​ParametricPlot3D[ℒ[k1],{v,-π,π},PlotStyle->{Blue,Thick}],ParametricPlot3D[
ℒ
1
[k1],{v,-π,π},PlotStyle->{Blue,Thick}],Graphics3D[{DarkGreen,Thick,Line[{{0,0,0},
ζ
11
[k1]}]}],Graphics3D[{DarkGreen,Thick,Line[{{0,0,0},
ζ
22
[k1]}]}]​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,1.},BoxRatiosAutomatic,PlotRange5,ImageSize900​​],{k1,-π,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
This clearly explains why the Pedal Surface it’s like above.