CuriousSurface2::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_]:={1.5(0.5+v-)Cos[u],1.5(0.5+v-)Sin[u],2v-1};[u_,v_]:=D[σ[u,v],u];[u_,v_]:=D[σ[u,v],v];p={0,0,0};==p+([u,v][u,v])//FullSimplify
3
v
3
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
σ
P
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
In[]:=
Manipulateσ[u_,v_]:={1.5(0.5+v-)Cos[u],1.5(0.5+v-)Sin[u],2v-1};[u_,v_]:=(27.`(0.6070072956246954`+1.`v)(0.8237133286601277`+v(-1.3570072956246952`+1.`v))Cos[u])(++),(27.`(0.6070072956246954`+1.`v)(0.8237133286601277`+v(-1.3570072956246952`+1.`v))Sin[u])(++),(60.75000000000004`(-0.5773502691896257`+1.`v)(0.5773502691896257`+1.`v)(0.6070072956246954`+1.`v)(0.8237133286601277`+v(-1.3570072956246952`+1.`v)))(++);ShowParametricPlot3D[σ[u,v],{u,0,2π},{v,-1,1.19},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.3]],ParametricPlot3D[u,v],{u,0,k1},v,-2,,PerformanceGoal->"Quality",PlotPoints60,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1],Graphics3D[{Black,Ball[{0,0,0},0.03]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{-10,10,5},PlotRangeAll,ImageSize500,{k1,0.00001,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
3
v
3
v
σ
P
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
σ
P
k1
π
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:
σ[u_,v_]:={1.5(0.5+v-)Cos[u],1.5(0.5+v-)Sin[u],2v-1};[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])//FullSimplify
3
v
3
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
σ
NP
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
In[]:=
Manipulateσ[u_,v_]:={1.5(0.5+v-)Cos[u],1.5(0.5+v-)Sin[u],2v-1};[u_,v_]:=-1.5`(-0.5`-1.`v+)-(27.`(0.6070072956246954`+1.`v)(0.8237133286601277`+v(-1.3570072956246952`+1.`v)))(++)Cos[u],-1.5`(-0.5`-1.`v+)-(27.`(0.6070072956246954`+1.`v)(0.8237133286601277`+v(-1.3570072956246952`+1.`v)))(++)Sin[u],-1+2v-(60.75000000000004`(-0.5773502691896257`+1.`v)(0.5773502691896257`+1.`v)(0.6070072956246954`+1.`v)(0.8237133286601277`+v(-1.3570072956246952`+1.`v)))(++);ShowParametricPlot3D[σ[u,v],{u,0,2π},{v,-1,1.19},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.1]],ParametricPlot3D[u,v],{u,0,k1},v,-1,,PerformanceGoal->"Quality",PlotPoints60,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8],Graphics3D[{Black,Ball[{0,0,0},0.03]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{-10,10,5},PlotRangeAll,ImageSize500,{k1,0.00001,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
3
v
3
v
σ
NP
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
2
(0.49999999999999994`+1.`v-1.`)
3
v
2
Abs[1.125`+v(2.25`+v(-3.375`-9.`v+6.75`))]
3
v
2
Abs[(1.5`+3.`v-3.`)Cos[u]]
3
v
2
Abs[(1.5`+3.`v-3.`)Sin[u]]
3
v
σ
NP
1.19k1
2π
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:
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).