CuriousSurface3::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+0.33vSinCos[u],1.5+0.33vSinSin[u],0.33vCos;[u_,v_]:=D[σ[u,v],u];[u_,v_]:=D[σ[u,v],v];p={0,0,0};==p+([u,v][u,v])//FullSimplify
u
2
u
2
u
2
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
u
2
u
2
u
2
3u
2
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
u
2
u
2
u
2
3u
2
5u
2
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
2
4.545454545454545`+1.`vSin+4.545454545454545`+1.`vSin
u
2
2
Cot[u]
u
2
5
Sin[u]
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
In[]:=
Manipulateσ[u_,v_]:=1.5+0.33vSinCos[u],1.5+0.33vSinSin[u],0.33vCos;[u_,v_]:=0.00444720375`Cot4.545454545454545`+1.`vSinSin[u]9.09090909090909`Cos[u]+1.`vSin+1.`vSin++,0.0088944075`Cot4.545454545454545`+1.`vSin2.2727272727272725`-2.2727272727272725`Cos[2u]+1.`vSin-0.25`vSin-0.25`vSin++,-0.0088944075`++;ShowParametricPlot3D[σ[u,v],{u,0,2π},{v,-1,1},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.3]],ParametricPlot3D[u,v],{u,0,k1},v,-2,,MaxRecursion->3,MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.3]],PlotStyleOpacity[0.8],Graphics3D[{Black,Ball[{0,0,0},0.04]}],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
u
2
u
2
u
2
σ
P
u
2
u
2
u
2
3u
2
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
u
2
u
2
u
2
3u
2
5u
2
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
2
4.545454545454545`+1.`vSin+4.545454545454545`+1.`vSin
u
2
2
Cot[u]
u
2
5
Sin[u]
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
σ
P
k1
π
Out[]=
Dark tone: Pedal surface. Light tone: Original 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_]:=1.5+0.33vSinCos[u],1.5+0.33vSinSin[u],0.33vCos;[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
u
2
u
2
u
2
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
u
2
u
2
u
2
u
2
3u
2
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
u
2
u
2
u
2
u
2
3u
2
5u
2
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
u
2
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
u
2
3u
2
2
Abs0.054450000000000005`vSin[u]+0.027225000000000003`v+0.12375`CscSin[2u]
u
2
u
2
2
AbsSin-0.495`-0.10890000000000001`vSin
u
2
u
2
2
Abs0.2475`Cos+0.2475`Cos+v(0.054450000000000005`+0.054450000000000005`Cos[u])Sin[u]
u
2
3u
2
2
Abs0.027225000000000006`v-0.054450000000000005`vCos[u]-0.027225000000000003`vCos[2u]+0.2475`Sin+0.2475`Sin
u
2
3u
2
Dark tone: Original surface. Light tone: Negative-pedal 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).