CuriousSurface5::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_]:=Sech[u]-Cos[v],Sin[v],
u
π
-Tanh[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

Cos[v]Sech[u](-1+π
2
Sech[u]
)(Cosh[u]-πSech[u]+Cos[v](-1+π-uTanh[u]))
2
Abs[Cos[v](-1+π
2
Sech[u]
)]
+
2
Abs[Sin[v]-π
2
Sech[u]
Sin[v]]
+
2
π
2
Abs[Cos[v]Sech[u]Tanh[u]]
,
(-1+π
2
Sech[u]
)Sin[v](-1+π
2
Sech[u]
+Cos[v]Sech[u](1-π+uTanh[u]))
2
Abs[Cos[v](-1+π
2
Sech[u]
)]
+
2
Abs[Sin[v]-π
2
Sech[u]
Sin[v]]
+
2
π
2
Abs[Cos[v]Sech[u]Tanh[u]]
,
πCos[v]Sech[u]Tanh[u](-1+π
2
Sech[u]
+Cos[v]Sech[u](1-π+uTanh[u]))
2
Abs[Cos[v](-1+π
2
Sech[u]
)]
+
2
Abs[Sin[v]-π
2
Sech[u]
Sin[v]]
+
2
π
2
Abs[Cos[v]Sech[u]Tanh[u]]
;
In[]:=
​​Manipulateσ[u_,v_]:=Sech[u]-Cos[v],Sin[v],
u
π
-Tanh[u];
σ
P
[u_,v_]:=
Cos[v]Sech[u](-1+π
2
Sech[u]
)(Cosh[u]-πSech[u]+Cos[v](-1+π-uTanh[u]))
2
Abs[Cos[v](-1+π
2
Sech[u]
)]
+
2
Abs[Sin[v]-π
2
Sech[u]
Sin[v]]
+
2
π
2
Abs[Cos[v]Sech[u]Tanh[u]]
,
(-1+π
2
Sech[u]
)Sin[v](-1+π
2
Sech[u]
+Cos[v]Sech[u](1-π+uTanh[u]))
2
Abs[Cos[v](-1+π
2
Sech[u]
)]
+
2
Abs[Sin[v]-π
2
Sech[u]
Sin[v]]
+
2
π
2
Abs[Cos[v]Sech[u]Tanh[u]]
,
πCos[v]Sech[u]Tanh[u](-1+π
2
Sech[u]
+Cos[v]Sech[u](1-π+uTanh[u]))
2
Abs[Cos[v](-1+π
2
Sech[u]
)]
+
2
Abs[Sin[v]-π
2
Sech[u]
Sin[v]]
+
2
π
2
Abs[Cos[v]Sech[u]Tanh[u]]
;​​ShowParametricPlot3D[σ[u,v],{u,-2π,2π},{v,-π,π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8]],​​ParametricPlot3D
σ
P
[u,v],{u,-2π,k1},v,-π,
k1
2
,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,-2.4,4.},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_]:=Sech[u]-Cos[v],Sin[v],
u
π
-Tanh[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[v]+Sech[u]--
Cos[v]
π
+Cos[v]
2
Sech[u]
(-Cos[v]+Sech[u])-
Cos[v]
π
+Cos[v]
2
Sech[u]
+Sin[v]
Sin[v]
π
-
2
Sech[u]
Sin[v]-Cos[v]Sech[u]
u
π
-Tanh[u]Tanh[u]
2
Abs-
Cos[v]
π
+Cos[v]
2
Sech[u]

+
2
Abs
Sin[v]
π
-
2
Sech[u]
Sin[v]
+
2
Abs[Cos[v]Sech[u]Tanh[u]]
,Sin[v]-
Sin[v]
π
-
2
Sech[u]
Sin[v](-Cos[v]+Sech[u])-
Cos[v]
π
+Cos[v]
2
Sech[u]
+Sin[v]
Sin[v]
π
-
2
Sech[u]
Sin[v]-Cos[v]Sech[u]
u
π
-Tanh[u]Tanh[u]
2
Abs-
Cos[v]
π
+Cos[v]
2
Sech[u]

+
2
Abs
Sin[v]
π
-
2
Sech[u]
Sin[v]
+
2
Abs[Cos[v]Sech[u]Tanh[u]]
,
u
π
-Tanh[u]+Cos[v]Sech[u]Tanh[u](-Cos[v]+Sech[u])-
Cos[v]
π
+Cos[v]
2
Sech[u]
+Sin[v]
Sin[v]
π
-
2
Sech[u]
Sin[v]-Cos[v]Sech[u]
u
π
-Tanh[u]Tanh[u]
2
Abs-
Cos[v]
π
+Cos[v]
2
Sech[u]

+
2
Abs
Sin[v]
π
-
2
Sech[u]
Sin[v]
+
2
Abs[Cos[v]Sech[u]Tanh[u]]
;
In[]:=
​​Manipulateσ[u_,v_]:=Sech[u]-Cos[v],Sin[v],
u
π
-Tanh[u];
σ
NP
[u_,v_]:=-Cos[v]+Sech[u]--
Cos[v]
π
+Cos[v]
2
Sech[u]
(-Cos[v]+Sech[u])-
Cos[v]
π
+Cos[v]
2
Sech[u]
+Sin[v]
Sin[v]
π
-
2
Sech[u]
Sin[v]-Cos[v]Sech[u]
u
π
-Tanh[u]Tanh[u]
2
Abs-
Cos[v]
π
+Cos[v]
2
Sech[u]

+
2
Abs
Sin[v]
π
-
2
Sech[u]
Sin[v]
+
2
Abs[Cos[v]Sech[u]Tanh[u]]
,Sin[v]-
Sin[v]
π
-
2
Sech[u]
Sin[v](-Cos[v]+Sech[u])-
Cos[v]
π
+Cos[v]
2
Sech[u]
+Sin[v]
Sin[v]
π
-
2
Sech[u]
Sin[v]-Cos[v]Sech[u]
u
π
-Tanh[u]Tanh[u]
2
Abs-
Cos[v]
π
+Cos[v]
2
Sech[u]

+
2
Abs
Sin[v]
π
-
2
Sech[u]
Sin[v]
+
2
Abs[Cos[v]Sech[u]Tanh[u]]
,
u
π
-Tanh[u]+Cos[v]Sech[u]Tanh[u](-Cos[v]+Sech[u])-
Cos[v]
π
+Cos[v]
2
Sech[u]
+Sin[v]
Sin[v]
π
-
2
Sech[u]
Sin[v]-Cos[v]Sech[u]
u
π
-Tanh[u]Tanh[u]
2
Abs-
Cos[v]
π
+Cos[v]
2
Sech[u]

+
2
Abs
Sin[v]
π
-
2
Sech[u]
Sin[v]
+
2
Abs[Cos[v]Sech[u]Tanh[u]]
;​​ShowParametricPlot3D[σ[u,v],{u,-2π,2π},{v,-π,π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1]],​​ParametricPlot3D
σ
NP
[u,v],{u,-2π,k1},v,-π,
k1
2
,MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[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,-2.4,4.},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:
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).