Negative-pedalsurfaceofanellipsoid::usage="Find the parametric equations of the negative-pedal surface of the ellipsoid ​
2
(x+3)
16
+
2
(y-4)
9
+
2
(z+1)
9
==1 regarding your center.Rederences: Personal, ResourceFunction: PedalSurface.";
Development:
Parametric equations of the negative-pedal surface of an ellipsoid:
σ[u_,v_]:={4Cos[u]Sin[v]-3,3Sin[u]Sin[v]+4,3Cos[v]-1};
σ
u
[u_,v_]:=D[σ[u,v],u];​​
σ
v
[u_,v_]:=D[σ[u,v],v];​​σ[u,v]-
(σ[u,v]-{-3,4,-1}).(
σ
u
[u,v]
σ
v
[u,v])
2
Norm[
σ
u
[u,v]
σ
v
[u,v]]
(
σ
u
[u,v]
σ
v
[u,v])//FullSimplify​​​​​​-3+4Cos[u]Sin[v]1-
9
2
Sin[v]
(9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
,
-48Sin[u]
3
Sin[v]
+9
2
Abs[Cos[u]]
4
Abs[Sin[v]]
(4+3Sin[u]Sin[v])+16
2
Abs[Sin[u]]
4
Abs[Sin[v]]
(4+3Sin[u]Sin[v])+4(4+3Sin[u]Sin[v])Sin[2v]Sin[2Conjugate[v]]
(9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
,-1+Cos[v]3-
48
2
Sin[v]
(9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
;
Another way to get it:
Plane parallel to the tangent that passes through the pedal point:
In[]:=
σ[u_,v_]:={4Cos[u]Sin[v]-3,3Sin[u]Sin[v]+4,3Cos[v]-1};
σ
u
[u_,v_]:=D[σ[u,v],u];​​
σ
v
[u_,v_]:=D[σ[u,v],v];N1=
σ
u
[u,v]
σ
v
[u,v]
Norm[
σ
u
[u,v]
σ
v
[u,v]]
//FullSimplify
In[]:=
-
3Cos[u]
2
Sin[v]
(9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
,-
4Sin[u]
2
Sin[v]
(9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
,-
2Sin[2v]
(9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
;​​{x,y,z}.N1-{-3,4,-1}.N1==0//FullSimplify
(3(3+x)Cos[u]+4(-4+y)Sin[u])
2
Sin[v]
+2(1+z)Sin[2v]0;​​
Normal line to the surface (does not pass through the pedal point):
x-(4Cos[u]Sin[v]-3)
3Cos[u]
2
Sin[v]
==
y-(3Sin[u]Sin[v]+4)
4Sin[u]
2
Sin[v]
==
z-(3Cos[v]-1)
2Sin[2v]
;
Parametric equations of the negative-pedal surface of an ellipsoid:
In[]:=
Solve(3(3+x)Cos[u]+4(-4+y)Sin[u])
2
Sin[v]
+2(1+z)Sin[2v]0,
x-(4Cos[u]Sin[v]-3)
3Cos[u]
2
Sin[v]
==
y-(3Sin[u]Sin[v]+4)
4Sin[u]
2
Sin[v]
==
z-(3Cos[v]-1)
2Sin[2v]
,{x,y,z}//FullSimplify
x
96+(75-21Cos[2u])
2
Tan[v]
-56Cos[u]Sin[v](1+
2
Sin[u]
2
Tan[v]
)
-32+(-25+7Cos[2u])
2
Tan[v]
,y
64
2
Csc[u]
+(64+36
2
Cot[u]
-21Cos[u]Cot[u]Sin[v])
2
Tan[v]
2
Csc[u]
(7+9
2
Sec[v]
)+7
2
Tan[v]
,z-1+
21Cos[v]
7-16
2
Csc[v]
2
Sec[u]
;
The graph is the same.
In[]:=
​​Manipulateσ[u_,v_]:={4Cos[u]Sin[v]-3,3Sin[u]Sin[v]+4,3Cos[v]-1};
σ
AP
[u_,v_]:=-3+4Cos[u]Sin[v]1-
9
2
Sin[v]
(9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
,(-48Sin[u]
3
Sin[v]
+9
2
Abs[Cos[u]]
4
Abs[Sin[v]]
(4+3Sin[u]Sin[v])+16
2
Abs[Sin[u]]
4
Abs[Sin[v]]
(4+3Sin[u]Sin[v])+4(4+3Sin[u]Sin[v])Sin[2v]Sin[2Conjugate[v]])((9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
),-1+Cos[v]3-
48
2
Sin[v]
(9
2
Abs[Cos[u]]
+16
2
Abs[Sin[u]]
)
4
Abs[Sin[v]]
+4
2
Abs[Sin[2v]]
;​​Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,0,π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],PlotStyleOpacity[0.1]],​​ParametricPlot3D[
σ
AP
[u,v],{u,0,k1},{v,0,k1},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],PlotStyleOpacity[0.6]],Graphics3D[{Black,Ball[{-3,4,-1},0.1]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{-10,10,5},PlotRangeAll,ImageSize500​​],{k1,0.00001,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
​
k1
Curve negative-pedal of the ellipse that results when cutting the ellipsoid with the plane z = - 1:
In[]:=
ζ[t_]:={4Cos[t]-3,3Sin[t]+4,-1};​​
ζ
11
[t_]={-3,4,-1}-
ζ'[t].({-3,4,-1}-ζ[t])
2
Norm[ζ'[t]]
ζ'[t]//FullSimplify
-3+
28
2
Sin[t]
9Cos[Conjugate[t]]+16Sin[Conjugate[t]]Tan[t]
,4-
21
2
Cos[t]
Sin[t]
9Cos[t]Cos[Conjugate[t]]+16Sin[t]Sin[Conjugate[t]]
,-1;
Animation of the ellipse negative-pedal curve:
In[]:=
​​SwatchLegend[{Black,Red,Blue,DarkGreen},{"Ellipse","Negative-pedal curve","Normal line to the ellipse","Orthogonal line to the normal line"}]​​Manipulateζ[t_]:={4Cos[t]-3,3Sin[t]+4,-1};
ζ
11
[t_]:=-3+
28
2
Sin[t]
9Cos[Conjugate[t]]+16Sin[Conjugate[t]]Tan[t]
,4-
21
2
Cos[t]
Sin[t]
9Cos[t]Cos[Conjugate[t]]+16Sin[t]Sin[Conjugate[t]]
,-1;​​Show[ParametricPlot3D[ζ[t],{t,0,2π},PlotStyle{Black,Thick}],​​Graphics3D[{Red,Ball[
ζ
11
[k1],0.1]}],Graphics3D[{Black,Ball[{-3,4,-1},0.1]}],​​ParametricPlot3D[
ζ
11
[t],{t,-π,π},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],​​Graphics3D[{Black,Ball[ζ[k1],0.1]}],​​Graphics3D[{Blue,Thick,InfiniteLine[{ζ[k1],
ζ
11
[k1]}]}],Graphics3D[{DarkGreen,Thick,InfiniteLine[{{-3,4,-1},
ζ
11
[k1]}]}]​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,20.},BoxRatiosAutomatic,PlotRange7,ImageSize900​​],{k1,-π,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
Ellipse
Negative-pedal curve
Normal line to the ellipse
Orthogonal line to the normal line
Out[]=
​
k1
This explains why the negative-pedal surface is there.