OffsetSurfaceSphere::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 sphere: {3 Cos[u]Cos[v],3 Sin[u]Cos[v],3 Sin[v]} with d = 2. References: Peternell, M. et al. Offsets, Conchoids and Pedal Surfaces; Personal. ";
σ
d
Development:
In[]:=
σ[u_,v_]:={3Cos[u]Cos[v],3Sin[u]Cos[v],3Sin[v]};[u_,v_]=D[σ[u,v],u];[u_,v_]=D[σ[u,v],v];d=2;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]3+++,Cos[v]3+++Sin[u],3+++Sin[v];
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
Cos[u]Cos[v]3-++,Cos[v]3-++Sin[u],3-++Sin[v];
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
Animation:
In[]:=
SwatchLegend[{Orange,Blue,Yellow},{"Positive Offset surface","Negative Offset surface","Base surface"}]Manipulateσ[u_,v_]:={3Cos[u]Cos[v],3Sin[u]Cos[v],3Sin[v]};[u_,v_]:=Cos[u]Cos[v]3+++,Cos[v]3+++Sin[u],3+++Sin[v];[u_,v_]:=Cos[u]Cos[v]3-++,Cos[v]3-++Sin[u],3-++Sin[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{{-7,7},{-7,7},{-5,5}},ImageSize700],{k1,0.00001,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
σ
dp
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
σ
dn
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
2Cos[v]
2
Abs[Cos[u]]
2
Cos[v]
2
Abs[Sin[u]]
2
Cos[v]
2
Abs[Cos[v]Sin[v]]
σ
dp
σ
dn
Out[]=
Out[]=
Explanation :
Cutting the surface:
2
x
2
y
2
z
with the plane y = 0, results:
2
x
2
z
In[]:=
SwatchLegend[{Orange,Blue,DarkYellow,Black},{"Positive Offset","Negative Offset","Original function","Generatrix line"}]Manipulateζ[t_]:={3Cos[t],0,3Sin[t]};d=2;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},{-5,5}},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
+