PedalSurfaceOfAtwoLeafHyperboloid::usage="Find the cartesian equation of the pedal surface of the two-leaf hyperboloid ​
2
z
-
2​
x
-
2
y
- 1 == 0, with respect to its center.Rederence: Personal.";
Development:
Equation of the plane tangent to the original surface at the point {x1, y1, z1}:
In[]:=
f[x_,y_,z_]:=
2
z
-
2
x
-
2
y
-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-x1)
f
x
[x1,y1,z1]+(y-y1)
f
y
[x1,y1,z1]+(z-z1)
f
z
[x1,y1,z1]==0​​
-2(x-x1)x1-2(y-y1)y1+2(z-z1)z10;​​-2xx1+2
2
x1
-2yy1+2
2
y1
+2zz1-2
2
z1
==0;​​-2x1x-2y1y+2z1z-2==0;​​x1x+y1y-z1z+1==0;
Coordinates of the tangent point:
In[]:=
Solvex1x+y1y-z1z+1==0,
x
x1
==
y
y1
==
z
-z1
,{x1,y1,z1}
x1-
x
2
x
+
2
y
+
2
z
,y1-
y
2
x
+
2
y
+
2
z
,z1
z
2
x
+
2
y
+
2
z
;
Cartesian equation of the pedal surface (replacing the coordinates of the tangent point in the equation of the original surface):
2
z
2
(
2
x
+
2
y
+
2
z
)
-
2
y
2
(
2
x
+
2
y
+
2
z
)
-
2
x
2
(
2
x
+
2
y
+
2
z
)
==1;​​
2
z
-
2
y
-
2
x
==
2
(
2
x
+
2
y
+
2
z
)
;​​
In[]:=
​​Manipulateσ[u_,v_]:=
2
u
-1
Cos[v],
2
u
-1
Sin[v],u;
σ
p
[x_,y_,z_]:=
2
z
-
2
y
-
2
x
-
2
(
2
x
+
2
y
+
2
z
)
;​​Show[ParametricPlot3D[σ[u,v],{u,-3,3},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],PlotStyleOpacity[0.1]],​​ContourPlot3D[
σ
p
[x,y,z]==0,{x,-1,k1},{y,-1,1},{z,-1,1},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],ContourStyleOpacity[0.6],PlotPoints->60],Graphics3D[{Black,Ball[{0,0,0},0.08]}],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​​],{k1,-1+0.00001,1,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
​
k1
To better understand why the pedal surface is like this, let’s make a cut with the plane y = 0:
In[]:=
2
z
-
2
x
==1;​​{Sinh[t],0,±Cosh[t]};​​ζ[t_]:={Sinh[t],0,Cosh[t]};​​
ζ
1
[t_]:={Sinh[t],0,-Cosh[t]};
Orthogonal projection of the point {0, 0, 0 } on the tangent line to the curve ζ[t] :
In[]:=
ζ'[t]
{Cosh[t],0,Sinh[t]};
Orthogonal plane to the tangent line (passes through the origin:
xCosh[t]+zSinh[t]==0;
Let’s get the coordinates of the orthogonally projected point (which will be the parametric equations of the pedal curve):
In[]:=
SolvexCosh[t]+zSinh[t]==0,
x-Sinh[t]
Cosh[t]
==
z-Cosh[t]
Sinh[t]
,y==0,{x,y,z}
x-
Cosh[t]Coth[t]-Sinh[t]
1+
2
Coth[t]
,y0,z
Coth[t](Cosh[t]Coth[t]-Sinh[t])
1+
2
Coth[t]
;
The parametric equations of the pedal curve ζ[t] would be:
ζ
p
[t_]:=-
Cosh[t]Coth[t]-Sinh[t]
1+
2
Coth[t]
,0,
Coth[t](Cosh[t]Coth[t]-Sinh[t])
1+
2
Coth[t]
;
Orthogonal projection of the point {0, 0, 0} On the tangent line to the curve
ζ
1
[t] :
In[]:=
ζ
1
'[t]
{Cosh[t],0,-Sinh[t]};
Orthogonal plane to the tangent line (passes through the origin):
xCosh[t]-zSinh[t]==0;
Let’s get the coordinates of the orthogonally projected point (which will be the parametric equations of the pedal curve):
In[]:=
SolvexCosh[t]-zSinh[t]==0,
x-Sinh[t]
Cosh[t]
==
z+Cosh[t]
-Sinh[t]
,y==0,{x,y,z}
x-
Cosh[t]Coth[t]-Sinh[t]
1+
2
Coth[t]
,y0,z-
Coth[t](Cosh[t]Coth[t]-Sinh[t])
1+
2
Coth[t]
;
The parametric equations of the pedal curve
ζ
1
[t] would be::
ζ
p1
[t_]:=-
Cosh[t]Coth[t]-Sinh[t]
1+
2
Coth[t]
,0,-
Coth[t](Cosh[t]Coth[t]-Sinh[t])
1+
2
Coth[t]
;
In[]:=
​​SwatchLegend[{Black,Red,Blue,DarkGreen},{"Branches of the hyperbola","Pedal curve to each branch of the hyperbola","Tangent line to each branch of the hyperbola","Orthogonal line to each tangent line"}]​​Manipulateζ[t_]:={Sinh[t],0,Cosh[t]};
ζ
1
[t_]:={Sinh[t],0,-Cosh[t]};
ζ
11
[t_]:=-
Cosh[t]Coth[t]-Sinh[t]
1+
2
Coth[t]
,0,
Coth[t](Cosh[t]Coth[t]-Sinh[t])
1+
2
Coth[t]
;​​
ζ
22
[t_]:=-
Cosh[t]Coth[t]-Sinh[t]
1+
2
Coth[t]
,0,-
Coth[t](Cosh[t]Coth[t]-Sinh[t])
1+
2
Coth[t]
;​​ℒ[t_]:=ζ[t]+vζ'[t];​​
ℒ
1
[t_]:=
ζ
1
[t]+v
ζ
1
'[t];​​​​Show[ParametricPlot3D[ζ[t],{t,-2,2},PlotStyle{Black,Thick}],ParametricPlot3D[
ζ
1
[t],{t,-2,2},PlotStyle{Black,Thick}],​​Graphics3D[{Black,Ball[ζ[k1],0.05]}],Graphics3D[{Black,Ball[
ζ
1
[k1],0.05]}],Graphics3D[{Black,Ball[{0,0,0},0.05]}],​​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],0.05]}],Graphics3D[{Red,Ball[
ζ
22
[k1],0.05]}],​​ParametricPlot3D[ℒ[k1],{v,-1.3,1.3},PlotStyle->{Blue,Thick}],ParametricPlot3D[
ℒ
1
[k1],{v,-1.3,1.3},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{0,-2,0},BoxRatiosAutomatic,PlotRange{{-3.5,4},{-4,4},{-3,3}},ImageSize900​​],{k1,-π,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
Branches of the hyperbola
Pedal curve to each branch of the hyperbola
Tangent line to each branch of the hyperbola
Orthogonal line to each tangent line
This clearly explains why the pedal surface is like this.