NegativePedalEllipticCylinder::usage="Find the parametric equations of the negative-pedal surface of the elliptic cylinder:​{3 Cos[u]​, 2 Sin[u]​, v} with respect to the origin.One way to obtain it is from its definition:​
σ
NP
== σ[u,v] - ​
σ[u,v]-

p
.(
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]​).​Reference: Personal.";
Development:
Negative-pedal surface:
In[]:=
σ[u_,v_]:={3Cos[u],2Sin[u],v};​​
σ
u
[u_,v_]=D[σ[u,v],u];​​
σ
v
[u_,v_]=D[σ[u,v],v];​​
σ
NP
==σ[u,v]-
σ[u,v].(
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​​​​
σ
NP

3(-4+4
2
Abs[Cos[u]]
+9
2
Abs[Sin[u]]
)Cos[u]
4Cos[u]Cos[Conjugate[u]]+9Sin[u]Sin[Conjugate[u]]
,
2(-9+4
2
Abs[Cos[u]]
+9
2
Abs[Sin[u]]
)Sin[u]
4Cos[u]Cos[Conjugate[u]]+9Sin[u]Sin[Conjugate[u]]
,v;
In[]:=
​​Manipulateσ[u_,v_]:={3Cos[u],2Sin[u],v};
σ
Np
[u_,v_]:=
3(-4+4
2
Abs[Cos[u]]
+9
2
Abs[Sin[u]]
)Cos[u]
4Cos[u]Cos[Conjugate[u]]+9Sin[u]Sin[Conjugate[u]]
,
2(-9+4
2
Abs[Cos[u]]
+9
2
Abs[Sin[u]]
)Sin[u]
4Cos[u]Cos[Conjugate[u]]+9Sin[u]Sin[Conjugate[u]]
,v;​​Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,-5,5},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],PlotStyle{Opacity[0.2],Orange}],​​ParametricPlot3D[
σ
Np
[u,v],{u,0,k1},{v,-5,5},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],PlotStyle{Opacity[0.8],Orange}],Graphics3D[{Black,Ball[{0,0,0},0.1]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,2.},PlotRangeAll,ImageSize500​​],{k1,0.00001,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
​
k1
The cylinder in ligth tone and its negative-pedal surface in dark tone.
Explanation:
To explain why the negative-pedal surface of a elliptic cylinder it is like this, we will make a cut with the plane z = 0 on the elliptic cylinder :
{3Cos[t],2Sin[t],0};
Negative-pedal curve:
In[]:=
ζ[t_]:={3Cos[t],2Sin[t],0};​​{0,0,0}-
ζ'[t].({0,0,0}-ζ[t])
2
Norm[ζ'[t]]
ζ'[t]//FullSimplify

15Cos[t]
2
Sin[t]
4Cos[t]Cos[Conjugate[t]]+9Sin[t]Sin[Conjugate[t]]
,-
10
2
Cos[t]
Sin[t]
4Cos[t]Cos[Conjugate[t]]+9Sin[t]Sin[Conjugate[t]]
,0;
In[]:=
​​SwatchLegend[{Black,Blue,DarkGreen,Red},{"Ellipse","Negative-pedal curve","Normal line to the ellipse","Orthogonal line to the normal line and the point {0, 0, 0}"}]​​Manipulateζ[t_]:={3Cos[t],2Sin[t],0};
ζ
NP
[t_]:=
15Cos[t]
2
Sin[t]
4Cos[t]Cos[Conjugate[t]]+9Sin[t]Sin[Conjugate[t]]
,-
10
2
Cos[t]
Sin[t]
4Cos[t]Cos[Conjugate[t]]+9Sin[t]Sin[Conjugate[t]]
,0;​​Show[{ParametricPlot3D[ζ[t],{t,0,2π},PlotStyle{Black,Thick}]/.Line[pts_,rest___]:>Tube[pts,0.035,rest],​​Graphics3D[{Blue,Ball[
ζ
NP
[k1],0.1]}],Graphics3D[{Red,Ball[{0,0,0},0.1]}],​​ParametricPlot3D[
ζ
NP
[t],{t,-π,k1},PlotStyle{Blue,Thick},PerformanceGoal->"Quality"]/.Line[pts_,rest___]:>Tube[pts,0.035,rest],​​Graphics3D[{Red,Thick,InfiniteLine[{{0,0,0},
ζ
NP
[k1]}]}]/.Line[pts_,rest___]:>Tube[pts,0.035,rest],Graphics3D[{DarkGreen,Thick,InfiniteLine[{ζ[k1],
ζ
NP
[k1]}]}]/.Line[pts_,rest___]:>Tube[pts,0.035,rest]}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-6,6},{-6,6},{-4,4}},ImageSize900,ViewPoint{1.3,-2.4,2.}​​],{k1,-π+0.00001,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
Ellipse
Negative-pedal curve
Normal line to the ellipse
Orthogonal line to the normal line and the point {0, 0, 0}
Out[]=
​
k1
That’s why the negative-pedal surface of an elliptic cylinder is that way .