SurfacesRelatedDEnneper::usage="It will be about obtaining a new surface from the original. Reference: Public domain; 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_]:=u-
3
u
3
+u
2
v
,v-
3
v
3
+v
2
u
,
2
u
-
2
v
;
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

2u(u-v)(u+v)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
,-
2(u-v)v(u+v)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
,
(u-v)(u+v)(-1+
2
u
+
2
v
)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
;
In[]:=
​​Manipulateσ[u_,v_]:=u-
3
u
3
+u
2
v
,v-
3
v
3
+v
2
u
,
2
u
-
2
v
;
σ
P
[u_,v_]:=
2u(u-v)(u+v)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
,-
2(u-v)v(u+v)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
,
(u-v)(u+v)(-1+
2
u
+
2
v
)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
;​​Show[ParametricPlot3D[σ[u,v],{u,-5,5},{v,-5,5},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.2]],​​ParametricPlot3D[
σ
P
[u,v],{u,-5,k1},{v,-5,k1},MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8]],Graphics3D[{Black,Ball[{0,0,0},0.2]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{-20,-10,-4.},PlotRange5,ImageSize500​​],{k1,-5+0.00001,5,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
​
k1
Dark tone: Pedal surface. Light tone: Original 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_]:=u-
3
u
3
+u
2
v
,v-
3
v
3
+v
2
u
,
2
u
-
2
v
;
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​​​​​​
σ
NP
u-
3
u
3
+u
2
v
-
2u(u-v)(u+v)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
,v+
2
u
v-
3
v
3
+
2(u-v)v(u+v)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
,
2
u
-
2
v
-
(u-v)(u+v)(-1+
2
u
+
2
v
)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
;
In[]:=
​​Manipulateσ[u_,v_]:=u-
3
u
3
+u
2
v
,v-
3
v
3
+v
2
u
,
2
u
-
2
v
;
σ
NP
[u_,v_]:=u-
3
u
3
+u
2
v
-
2u(u-v)(u+v)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
,v+
2
u
v-
3
v
3
+
2(u-v)v(u+v)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
,
2
u
-
2
v
-
(u-v)(u+v)(-1+
2
u
+
2
v
)
2
(1+
2
u
+
2
v
)
(3+
2
u
+
2
v
)
3(4(
2
Abs[u]
+
2
Abs[v]
)
2
Abs[1+
2
u
+
2
v
]
+
2
Abs[-1+
2
(
2
u
+
2
v
)
]
)
;​​Show[ParametricPlot3D[σ[u,v],{u,-5,5},{v,-5,5},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.2]],​​ParametricPlot3D[
σ
NP
[u,v],{u,-5,k1},{v,-5,k1},MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8]],Graphics3D[{Black,Ball[{0,0,0},0.2]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{-20,-10,-4.},PlotRange5,ImageSize500​​],{k1,-5+0.00001,5,0.00001},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_]:=u-
3
u
3
+u
2
v
,v-
3
v
3
+v
2
u
,
2
u
-
2
v
;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: Conchoid surfaces (positive and negative). Light tone: Original surface.
​
​
Development of Offset Surfaces of the original:
Parametric equations of the offset surfaces of the original surface :
Dark tone: Offset surfaces (positive and negative). Light tone: Original surface.