OffsetSurfaceEllipsoid::usage="An Offset surface, is a smooth surface (without edges, peaks, or singular points, where each point has a tangent plane. Examples: Spheres, ellipsoids, toroids, cylinders and planes), it is defined as follows: [u, v] = σ[u, v] ± d n[u, v] σ[u, v] it is a smooth surface, d = distance, n[u, v] it is the normal unitary vector to the surface σ[u, v]. In this notebook we will obtain the offset surface of the ellipsoid: {4 Cos[u]Cos[v],3 Sin[u]Cos[v],2 Sin[v]}. References: Peternell, M. et al. Offsets, Conchoids and Pedal Surfaces; Personal. ";
σ
d
Development:
In[]:=
σ[u_,v_]:={4Cos[u]Cos[v],3Sin[u]Cos[v],2Sin[v]};[u_,v_]=D[σ[u,v],u];[u_,v_]=D[σ[u,v],v];d=1.5;n[u_,v_]:=[u,v][u,v];σ[u,v]+dn[u,v]//FullSimplifyσ[u,v]-dn[u,v]//FullSimplify
σ
u
σ
v
σ
u
σ
v
Norm[[u,v][u,v]]
σ
u
σ
v
Cos[u]Cos[v]4+(9+16)+9,Cos[v]3+(9+16)+9Sin[u],2+(9+16)+9Sin[v];
4.5`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
6.`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
9.`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
Cos[u]Cos[v]4-(9+16)+9,Cos[v]3-(9+16)+9Sin[u],2-(9+16)+9Sin[v];
4.5`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
6.`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
9.`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
Animation:
SwatchLegend[{Orange,Blue,Yellow},{"Positive Offset surface","Negative Offset surface","Base surface"}]Manipulateσ[u_,v_]:={4Cos[u]Cos[v],3Sin[u]Cos[v],2Sin[v]};[u_,v_]:=Cos[u]Cos[v]4+(9+16)+9,Cos[v]3+(9+16)+9Sin[u],2+(9+16)+9Sin[v];[u_,v_]:=Cos[u]Cos[v]4-(9+16)+9,Cos[v]3-(9+16)+9Sin[u],2-(9+16)+9Sin[v];Show[ParametricPlot3D[σ[u,v],{u,0,π},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],PlotStyle{Opacity[0.4],Yellow},Mesh->False,PlotPoints60],ParametricPlot3D[[u,v],{u,0,k1},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],PlotStyle{Opacity[0.2],Orange},PlotPoints60],ParametricPlot3D[[u,v],{u,0,k1},{v,0,2π},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],PlotStyle{Opacity[0.2],Blue},PlotPoints60],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,2.},PlotLabelStyle[Framed["Offset surfaces"],16,Blue,BackgroundYellow],PlotRange{{-6,6},{-6,6},{-4,4}},ImageSize700],{k1,0.00001,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
σ
dp
4.5`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
6.`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
9.`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
σ
dn
4.5`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
6.`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
9.`Cos[v]
4
Abs[Cos[v]]
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Sin[2v]]
σ
dp
σ
dn
Out[]=
Out[]=
Explanation 1:
Cutting the surface:
2
x
16
2
y
9
2
z
4
with the plane y = 0, results:
2
x
16
2
z
4
In[]:=
SwatchLegend[{Orange,Blue,DarkYellow,Black},{"Positive Offset","Negative Offset","Original function","Generatrix line"}]Manipulateζ[t_]:={4Cos[t],0,2Sin[t]};d=1.5;N1[t_]:=;[t_]:=ζ[t]+dN1[t];[t_]:=ζ[t]-dN1[t];;Show[{ParametricPlot3D[[t],{t,0,t1},PlotStyle{Orange,Thick},PerformanceGoal->"Quality",PlotPoints60],ParametricPlot3D[[t],{t,0,t1},PlotStyle{Blue,Thick},PerformanceGoal->"Quality",PlotPoints60],ParametricPlot3D[ζ[t],{t,0,2π},PlotStyle{DarkYellow,Thick}],Graphics3D[{Orange,Ball[[t1],0.1]}],Graphics3D[{Blue,Ball[[t1],0.1]}],Graphics3D[{Black,Thick,Line[{[t1],ζ[t1],[t1]}]}],Graphics3D[{DarkYellow,Ball[ζ[t1],0.1]}]},AxesTrue,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),BoxedFalse,AxesOrigin{0,0,0},BoxRatiosAutomatic,ImageSize700,PlotRange{{-6,6},{-6,6},{-4,4}},ViewPoint{1.3,-2.4,2.},PlotLabelStyle[Framed["Offset curves"],16,Blue,BackgroundYellow]],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacement->Top
(ζ'[t]ζ''[t])ζ'[t]
Norm[ζ'[t]ζ''[t]]Norm[ζ'[t]]
ζ
+
ζ
-
ζ
+
ζ
-
ζ
+
ζ
-
ζ
-
ζ
+
Explanation 2:
Cutting the surface:
with the plane x = 0, results: