PedalNegativePedalToroid::usage="Find the parametric equations of the pedal and negative-pedal surfaces of the toroid:{(4 + 2 Cos[u])Cos[v], (4 + 2 Cos[u])Sin[v], 2 Sin[u]} with respect to the origin.One way to obtain it is from its definitions: == + ([u,v][u, v]). == σ[u,v] - ([u,v][u, v]).Reference: Personal";
σ
p
p
σ[u,v]-.([u,v][u,v])
p
σ
u
σ
v
2
Norm[[u,v][u,v]]
σ
u
σ
v
σ
u
σ
v
σ
NP
σ[u,v]-.([u,v][u,v])
p
σ
u
σ
v
2
Norm[[u,v][u,v]]
σ
u
σ
v
σ
u
σ
v
Development:
Pedal surface:
In[]:=
σ[u_,v_]:={(4+2Cos[u])Cos[v],(4+2Cos[u])Sin[v],2Sin[u]};[u_,v_]=D[σ[u,v],u];[u_,v_]=D[σ[u,v],v];==([u,v][u,v])//FullSimplify
σ
u
σ
v
σ
p
σ[u,v].([u,v][u,v])
σ
u
σ
v
2
Norm[[u,v][u,v]]
σ
u
σ
v
σ
u
σ
v
σ
p
2(2+Cos[u])(1+Cos[u]+Cos[2u])Cos[v]
(2+Cos[Conjugate[u]])(Cos[u]Cos[Conjugate[u]]Cosh[2Im[v]]+Sin[u]Sin[Conjugate[u]])
2(2+Cos[u])(1+Cos[u]+Cos[2u])Sin[v]
(2+Cos[Conjugate[u]])(Cos[u]Cos[Conjugate[u]]Cosh[2Im[v]]+Sin[u]Sin[Conjugate[u]])
2(2+Cos[u])(1+2Cos[u])Sin[u]
(2+Cos[Conjugate[u]])(Cos[u]Cos[Conjugate[u]]Cosh[2Im[v]]+Sin[u]Sin[Conjugate[u]])
Manipulateσ[u_,v_]:={(4+2Cos[u])Cos[v],(4+2Cos[u])Sin[v],2Sin[u]};[u_,v_]:=,,;Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],PlotStyleOpacity[0.4]],ParametricPlot3D[[u,v],{u,0,k1},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],PlotStyleOpacity[0.2]],Graphics3D[{Black,Ball[{0,0,0},0.1]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{-10,10,10},PlotRangeAll,ImageSize500],{k1,0.00001,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
σ
p
2(2+Cos[u])(1+Cos[u]+Cos[2u])Cos[v]
(2+Cos[Conjugate[u]])(Cos[u]Cos[Conjugate[u]]Cosh[2Im[v]]+Sin[u]Sin[Conjugate[u]])
2(2+Cos[u])(1+Cos[u]+Cos[2u])Sin[v]
(2+Cos[Conjugate[u]])(Cos[u]Cos[Conjugate[u]]Cosh[2Im[v]]+Sin[u]Sin[Conjugate[u]])
2(2+Cos[u])(1+2Cos[u])Sin[u]
(2+Cos[Conjugate[u]])(Cos[u]Cos[Conjugate[u]]Cosh[2Im[v]]+Sin[u]Sin[Conjugate[u]])
σ
p
Out[]=
The toroid in dark tone and its pedal surface in ligth tone.
Negative-pedal surface:
In[]:=
σ[u_,v_]:={(4+2Cos[u])Cos[v],(4+2Cos[u])Sin[v],2Sin[u]};[u_,v_]=D[σ[u,v],u];[u_,v_]=D[σ[u,v],v];==σ[u,v]-([u,v][u,v])//FullSimplify
σ
u
σ
v
σ
NP
σ[u,v].([u,v][u,v])
σ
u
σ
v
2
Norm[[u,v][u,v]]
σ
u
σ
v
σ
u
σ
v
σ
NP
2(2+Cos[u])(++-Cos[u](2+Cos[u])(1+2Cos[u]))Cos[v]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2(2+Cos[u])(++-Cos[u](2+Cos[u])(1+2Cos[u]))Sin[v]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2(++-(1+2Cos[u]))Sin[u]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2
(2+Cos[u])
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
In[]:=
Manipulateσ[u_,v_]:={(4+2Cos[u])Cos[v],(4+2Cos[u])Sin[v],2Sin[u]};[u_,v_]:=++,++,++;Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],PlotStyleOpacity[0.2]],ParametricPlot3D[[u,v],{u,0,k1},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],PlotStyleOpacity[0.6]],Graphics3D[{Black,Ball[{0,0,0},0.1]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{-10,10,10},PlotRangeAll,ImageSize500],{k1,0.00001,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
σ
Np
2(2+Cos[u])(++-Cos[u](2+Cos[u])(1+2Cos[u]))Cos[v]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2(2+Cos[u])(++-Cos[u](2+Cos[u])(1+2Cos[u]))Sin[v]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2(++-(1+2Cos[u]))Sin[u]
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
2
(2+Cos[u])
2
Abs[Cos[u](2+Cos[u])Cos[v]]
2
Abs[(2+Cos[u])Sin[u]]
2
Abs[Cos[u](2+Cos[u])Sin[v]]
σ
Np
Out[]=
The toroid in ligth tone and its negative-pedal surface in dark tone.
Explanation:
To explain why the pedal and negative-pedal surfaces of a toroid are like this, we will make a cut with the plane x = 0 on the toroid and there are two circumferences:
In 3D and with parametric equations the circumferences would be:
Pedal curves:
That’s why the pedal surface of a toroid is that way (visualize in 3D)
Negative-pedal curves:
That’s why the negative-pedal surface of a toroid is that way (visualize in 3D)