CuriousSurface1::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_]:={Sin[2u]Cos[v],Sin[2u]Sin[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])//FullSimplify​​​​​​
σ
P

8
3
Cos[u]
Cos[v]
5
Sin[u]
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
,
8
3
Cos[u]
5
Sin[u]
Sin[v]
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
,-
2
3
Sin[u]
Sin[2u]Sin[4u]
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
;
In[]:=
​​Manipulateσ[u_,v_]:={Sin[2u]Cos[v],Sin[2u]Sin[v],Sin[u]};
σ
P
[u_,v_]:=
8
3
Cos[u]
Cos[v]
5
Sin[u]
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
,
8
3
Cos[u]
5
Sin[u]
Sin[v]
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
,-
2
3
Sin[u]
Sin[2u]Sin[4u]
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
;​​Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,0,π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8]],​​ParametricPlot3D[
σ
P
[u,v],{u,0,k1},{v,0,k1},PerformanceGoal->"Quality",PlotPoints60,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1]],Graphics3D[{Black,Ball[{0,0,0},0.02]}],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
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_]:={Sin[2u]Cos[v],Sin[2u]Sin[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])//FullSimplify//FullSimplify​​​​​​
σ
NP
(Cos[v](
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
-4
2
Cos[u]
4
Sin[u]
)Sin[2u])(
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
),((
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
-4
2
Cos[u]
4
Sin[u]
)Sin[2u]Sin[v])(
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
),Sin[u]+
2
3
Sin[u]
Sin[2u]Sin[4u]
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
;
In[]:=
​​Manipulateσ[u_,v_]:={Sin[2u]Cos[v],Sin[2u]Sin[v],Sin[u]};
σ
NP
[u_,v_]:=(Cos[v](
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
-4
2
Cos[u]
4
Sin[u]
)Sin[2u])(
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
),((
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
-4
2
Cos[u]
4
Sin[u]
)Sin[2u]Sin[v])(
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
),Sin[u]+
2
3
Sin[u]
Sin[2u]Sin[4u]
2
Abs[Cos[u]Cos[v]Sin[2u]]
+
2
Abs[Sin[4u]]
+
2
Abs[Cos[u]Sin[2u]Sin[v]]
;​​Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,0,π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1]],​​ParametricPlot3D[
σ
NP
[u,v],{u,0,k1},{v,0,k1},PerformanceGoal->"Quality",PlotPoints60,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8]],Graphics3D[{Black,Ball[{0,0,0},0.02]}],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
Dark tone: Negative-pedal surface. Light tone: Original surface.
​
​
Development of Conchoid Surfaces of the original:
Parametric equations of the conchoid surfaces of the original surface with respect to the origin:
In[]:=
σ[u_,v_]:={Sin[2u]Cos[v],Sin[2u]Sin[v],Sin[u]};k=3;​​p={0,0,0};​​
σ
c1
==σ[u,v]+k
σ[u,v]+p
Norm[σ[u,v]-p]
//FullSimplify​​
σ
c2
==σ[u,v]-k
σ[u,v]+p
Norm[σ[u,v]-p]
//FullSimplify​​
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).