CuriousSurface7::usage="It will be about obtaining a new surface from the original. Reference: Internet: proyectodescartes.org; Personal.";
Development of Pedal Surface of the original:
Parametric equations of the pedal surface of the original surface with respect to the origin:
In[]:=
σ[u_,v_]:={Cos[v]Cos[u],-Sin[v]Cos[u],v+Sin[u]};
σ
u
[u_,v_]:=D[σ[u,v],u];​​
σ
v
[u_,v_]:=D[σ[u,v],v];p={0,0,0};​​
σ
P
==p+
(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])​​​​​​
σ
P
((
2
Cos[u]
Cos[v]+Sin[u]Sin[v])(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
),((Cos[v]Sin[u]-
2
Cos[u]
Sin[v])(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
),((Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
);
In[]:=
​​Manipulateσ[u_,v_]:={Cos[v]Cos[u],-Sin[v]Cos[u],v+Sin[u]};
σ
P
[u_,v_]:=((
2
Cos[u]
Cos[v]+Sin[u]Sin[v])(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
),((Cos[v]Sin[u]-
2
Cos[u]
Sin[v])(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
),((Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
);​​Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.6]],​​ParametricPlot3D[
σ
P
[u,v],{u,0,k1},{v,0,k1},MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1]],Graphics3D[{Black,Ball[{0,0,0},0.04]}],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
Dark tone: Original surface. Light tone: Pedal surface
​
​
Development of Negative-Pedal Surface of the original:
Parametric equations of the negative-pedal surface of the original surface with respect to the origin:
In[]:=
σ[u_,v_]:={Cos[v]Cos[u],-Sin[v]Cos[u],v+Sin[u]};
σ
u
[u_,v_]:=D[σ[u,v],u];​​
σ
v
[u_,v_]:=D[σ[u,v],v];p={0,0,0};​​
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])​​​​​​
σ
NP
Cos[u]Cos[v]-((
2
Cos[u]
Cos[v]+Sin[u]Sin[v])(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
),-Cos[u]Sin[v]-((Cos[v]Sin[u]-
2
Cos[u]
Sin[v])(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
),v+Sin[u]-((Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
);
In[]:=
​​Manipulateσ[u_,v_]:={Cos[v]Cos[u],-Sin[v]Cos[u],v+Sin[u]};
σ
NP
[u_,v_]:=Cos[u]Cos[v]-((
2
Cos[u]
Cos[v]+Sin[u]Sin[v])(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
),-Cos[u]Sin[v]-((Cos[v]Sin[u]-
2
Cos[u]
Sin[v])(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
),v+Sin[u]-((Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)(-Cos[u]Sin[v](Cos[v]Sin[u]-
2
Cos[u]
Sin[v])+Cos[u]Cos[v](
2
Cos[u]
Cos[v]+Sin[u]Sin[v])+(v+Sin[u])(Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
)))(
2
Abs[Cos[v]Sin[u]-
2
Cos[u]
Sin[v]]
+
2
Abs[
2
Cos[u]
Cos[v]+Sin[u]Sin[v]]
+
2
Abs[Cos[u]
2
Cos[v]
Sin[u]+Cos[u]Sin[u]
2
Sin[v]
]
);​​Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8]],​​ParametricPlot3D[
σ
NP
[u,v],{u,0,k1},{v,0,k1},MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1]],Graphics3D[{Black,Ball[{0,0,0},0.04]}],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
Dark tone: Original surface. Light tone: Negative-pedal surface.
​
​
Development of Conchoid Surfaces of the original:
Parametric equations of the conchoid surfaces of the original surface with respect to the origin:
Dark tone: Original surface. Light tone: Conchoid surfaces (positive and negative).
​
​
Development of Offset Surfaces of the original:
Parametric equations of the offset surfaces of the original surface :
Dark tone: Original surface. Light tone: Offset surfaces (positive and negative).