CuriousSurface4::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_]:=(2.2+1.05Cos[v])Cos
u
2
,2.2+1.05Cos[v]Cos
u
2
Sin[u],1.05Sin[v]Cos
u
2
;
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])//FullSimplify​​​​​​
σ
P
1.2762815625000001`
2
Cos
u
2

Cos[u]Cos[v]2.095238095238096`+1.`Cos
u
2
Cos[v]-0.5`Sin
u
2
Sin[u]
2
Cos
u
2

Cos[u]Cos[v](1.`+2.0952380952380953`Cos[v])+Cos
u
2
Cos[u](2.0952380952380953`+4.3900226757369625`Cos[v])+(1.0476190476190477`+2.1950113378684812`Cos[v])Sin
u
2
Sin[u]0.33350625000000017`
2
Abs[(0.4772727272727272`+1.`Cos[v])Sin[u]]
+
2
AbsCos
u
2
Cos[u]Cos[v]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]-0.55125`Sin
u
2
Sin[u]
+
2
AbsCos
u
2
Cos[u]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]+1.1550000000000002`Sin
u
2
Sin[u]Sin[v]
,0.6381407812500001`
2
Cos
u
2

(1.`+2.095238095238096`Cos[v])Sin
u
2

2
Cos
u
2

Cos[u]Cos[v](1.`+2.0952380952380953`Cos[v])+Cos
u
2
Cos[u](2.0952380952380953`+4.3900226757369625`Cos[v])+(1.0476190476190477`+2.1950113378684812`Cos[v])Sin
u
2
Sin[u]0.33350625000000017`
2
Abs[(0.4772727272727272`+1.`Cos[v])Sin[u]]
+
2
AbsCos
u
2
Cos[u]Cos[v]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]-0.55125`Sin
u
2
Sin[u]
+
2
AbsCos
u
2
Cos[u]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]+1.1550000000000002`Sin
u
2
Sin[u]Sin[v]
,1.2762815625000001`
2
Cos
u
2

Cos[u]2.095238095238096`+1.`Cos
u
2
Cos[v]+1.047619047619048`Sin
u
2
Sin[u]
2
Cos
u
2

Cos[u]Cos[v](1.`+2.0952380952380953`Cos[v])+Cos
u
2
Cos[u](2.0952380952380953`+4.3900226757369625`Cos[v])+(1.0476190476190477`+2.1950113378684812`Cos[v])Sin
u
2
Sin[u]Sin[v]0.33350625000000017`
2
Abs[(0.4772727272727272`+1.`Cos[v])Sin[u]]
+
2
AbsCos
u
2
Cos[u]Cos[v]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]-0.55125`Sin
u
2
Sin[u]
+
2
AbsCos
u
2
Cos[u]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]+1.1550000000000002`Sin
u
2
Sin[u]Sin[v]
;
In[]:=
​​Manipulateσ[u_,v_]:=(2.2+1.05Cos[v])Cos
u
2
,2.2+1.05Cos[v]Cos
u
2
Sin[u],1.05Sin[v]Cos
u
2
;
σ
P
[u_,v_]:=1.2762815625000001`
2
Cos
u
2

Cos[u]Cos[v]2.095238095238096`+1.`Cos
u
2
Cos[v]-0.5`Sin
u
2
Sin[u]
2
Cos
u
2

Cos[u]Cos[v](1.`+2.0952380952380953`Cos[v])+Cos
u
2
Cos[u](2.0952380952380953`+4.3900226757369625`Cos[v])+(1.0476190476190477`+2.1950113378684812`Cos[v])Sin
u
2
Sin[u]0.33350625000000017`
2
Abs[(0.4772727272727272`+1.`Cos[v])Sin[u]]
+
2
AbsCos
u
2
Cos[u]Cos[v]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]-0.55125`Sin
u
2
Sin[u]
+
2
AbsCos
u
2
Cos[u]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]+1.1550000000000002`Sin
u
2
Sin[u]Sin[v]
,0.6381407812500001`
2
Cos
u
2

(1.`+2.095238095238096`Cos[v])Sin
u
2

2
Cos
u
2

Cos[u]Cos[v](1.`+2.0952380952380953`Cos[v])+Cos
u
2
Cos[u](2.0952380952380953`+4.3900226757369625`Cos[v])+(1.0476190476190477`+2.1950113378684812`Cos[v])Sin
u
2
Sin[u]0.33350625000000017`
2
Abs[(0.4772727272727272`+1.`Cos[v])Sin[u]]
+
2
AbsCos
u
2
Cos[u]Cos[v]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]-0.55125`Sin
u
2
Sin[u]
+
2
AbsCos
u
2
Cos[u]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]+1.1550000000000002`Sin
u
2
Sin[u]Sin[v]
,1.2762815625000001`
2
Cos
u
2

Cos[u]2.095238095238096`+1.`Cos
u
2
Cos[v]+1.047619047619048`Sin
u
2
Sin[u]
2
Cos
u
2

Cos[u]Cos[v](1.`+2.0952380952380953`Cos[v])+Cos
u
2
Cos[u](2.0952380952380953`+4.3900226757369625`Cos[v])+(1.0476190476190477`+2.1950113378684812`Cos[v])Sin
u
2
Sin[u]Sin[v]0.33350625000000017`
2
Abs[(0.4772727272727272`+1.`Cos[v])Sin[u]]
+
2
AbsCos
u
2
Cos[u]Cos[v]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]-0.55125`Sin
u
2
Sin[u]
+
2
AbsCos
u
2
Cos[u]2.3100000000000005`+1.1025`Cos
u
2
Cos[v]+1.1550000000000002`Sin
u
2
Sin[u]Sin[v]
;​​ShowParametricPlot3D[σ[u,v],{u,-π,π},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8]],​​ParametricPlot3D
σ
P
[u,v],u,-π,
k1
2
,{v,0,k1},MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1],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,-2.4,2.},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_]:=(2.2+1.05Cos[v])Cos
u
2
,2.2+1.05Cos[v]Cos
u
2
Sin[u],1.05Sin[v]Cos
u
2
;
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])​​​​​​
In[]:=
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).