CuriousSurface6::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_]:={0.5(1-Cos[v])Sin[v]Cos[u],0.5(1-Cos[v])Sin[v]Sin[u],Cos[v]};[u_,v_]:=D[σ[u,v],u];[u_,v_]:=D[σ[u,v],v];p={0,0,0};==p+([u,v][u,v])
σ
u
σ
v
σ
P
(σ[u,v]-p).([u,v][u,v])
σ
u
σ
v
2
Norm[[u,v][u,v]]
σ
u
σ
v
σ
u
σ
v
In[]:=
Out[]=
Dark tone: Original surface. Light tone: Pedal surface
Development of Negative-Pedal Surface of the original:
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_]:={0.5(1-Cos[v])Sin[v]Cos[u],0.5(1-Cos[v])Sin[v]Sin[u],Cos[v]};[u_,v_]:=D[σ[u,v],u];[u_,v_]:=D[σ[u,v],v];p={0,0,0};==σ[u,v]-([u,v][u,v])
σ
u
σ
v
σ
NP
(σ[u,v]-p).([u,v][u,v])
σ
u
σ
v
2
Norm[[u,v][u,v]]
σ
u
σ
v
σ
u
σ
v
In[]:=
Out[]=
Dark tone: Negative-pedal surface. Light tone: Original surface.
Development of Conchoid Surfaces of the original:
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_]:={0.5(1-Cos[v])Sin[v]Cos[u],0.5(1-Cos[v])Sin[v]Sin[u],Cos[v]};k=3;p={0,0,0};==σ[u,v]+k//FullSimplify==σ[u,v]-k//FullSimplify
σ
c1
σ[u,v]+p
Norm[σ[u,v]-p]
σ
c2
σ[u,v]+p
Norm[σ[u,v]-p]
σ
c1
1.5`
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
1.5`
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
3
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
σ
c2
1.5`
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
1.5`
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
3
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
In[]:=
Manipulateσ[u_,v_]:={0.5(1-Cos[v])Sin[v]Cos[u],0.5(1-Cos[v])Sin[v]Sin[u],Cos[v]};[u_,v_]:=0.5`++0.25`+0.25`Cos[u](1-Cos[v])Sin[v],0.5`++0.25`+0.25`(1-Cos[v])Sin[u]Sin[v],1++0.25`+0.25`Cos[v];[u_,v_]:=0.5`-+0.25`+0.25`Cos[u](1-Cos[v])Sin[v],0.5`-+0.25`+0.25`(1-Cos[v])Sin[u]Sin[v],1-+0.25`+0.25`Cos[v];ShowParametricPlot3D[σ[u,v],{u,0,π},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.6]],ParametricPlot3D[u,v],u,0,,{v,0,k1},MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1],ParametricPlot3D[u,v],u,0,,{v,0,k1},MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1],Graphics3D[{Black,Ball[{0,0,0},0.04]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,2.},PlotRangeAll,ImageSize500,{k1,0.00001,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
σ
c1
1.5`
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
1.5`
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
3
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
σ
c2
1.5`
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
1.5`
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
3
2
Abs[Cos[v]]
2
Abs[Cos[u](-1+Cos[v])Sin[v]]
2
Abs[(-1+Cos[v])Sin[u]Sin[v]]
σ
c1
k1
2
σ
c2
k1
2
Dark tone: Original surface. Light tone: Conchoid surfaces (positive and negative).
Development of Offset Surfaces of the original:
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).