Wolfram Cloud Document

OffsetSurfaceCylinder::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 cylinder: {Cos[u],Sin[u],v} with d = 1.5. References: Peternell, M. et al. Offsets, Conchoids and Pedal Surfaces; Personal. ";

Development:

In[]:=

σ[u_,v_]:={Cos[u],Sin[u],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]

Norm[

[u,v]

[u,v]]

;σ[u,v]+dn[u,v]//FullSimplifyσ[u,v]-dn[u,v]//FullSimplify

1+

1.5`

Abs[Cos[u]]

Abs[Sin[u]]

Cos[u],1+

1.5`

Abs[Cos[u]]

Abs[Sin[u]]

Sin[u],0.`+v;

1-

1.5`

Abs[Cos[u]]

Abs[Sin[u]]

Cos[u],1-

1.5`

Abs[Cos[u]]

Abs[Sin[u]]

Sin[u],0.`+v;

Animation:

In[]:=

SwatchLegend[{Orange,Blue,Yellow},{"Positive Offset surface","Negative Offset surface","Base surface"}]Manipulate[σ[u_,v_]:={Cos[u],Sin[u],v};

[u_,v_]:={(1+1.5`)Cos[u],(1+1.5`)Sin[u],0.`+v};

[u_,v_]:={(1-1.5`)Cos[u],(1-1.5`)Sin[u],0.`+v};Show[ParametricPlot3D[σ[u,v],{u,0,2π},{v,-5,5},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],PlotStyle{Opacity[0.8],Yellow},Mesh->False,PlotPoints60],ParametricPlot3D[

[u,v],{u,0,k1},{v,-5,5},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],PlotStyle{Opacity[0.4],Orange},PlotPoints60],ParametricPlot3D[

[u,v],{u,0,k1},{v,-5,5},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.4]],PlotStyle{Opacity[0.4],Blue},PlotPoints60],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,2.},PlotLabelStyle[Framed["Offset surfaces"],16,Blue,BackgroundYellow],PlotRange{{-3,3},{-3,3},{-5,5}},ImageSize700],{k1,0.00001,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop]

Out[]=

	Positive Offset surface
	Negative Offset surface
	Base surface

Out[]=

Explanation :

In[]:=

Eliminate[{x==Cos[u],y==Sin[u],z==v},{u,v}]



;

Cutting the surface:

==1;

with the plane z = 1, results:

{Cos[t],Sin[t],1};

In[]:=

SwatchLegend[{Orange,Blue,DarkYellow,Black},{"Positive Offset","Negative Offset","Original function","Generatrix line"}]Manipulateζ[t_]:={Cos[t],Sin[t],1};d=1.5;N1[t_]:=

(ζ'[t]ζ''[t])ζ'[t]

Norm[ζ'[t]ζ''[t]]Norm[ζ'[t]]

;

[t_]:=ζ[t]+dN1[t];

[t_]:=ζ[t]-dN1[t];;Show[{ParametricPlot3D[

[t],{t,0,t1},PlotStyle{Orange,Thick},PerformanceGoal->"Quality",PlotPoints60],ParametricPlot3D[

[t],{t,0,t1},PlotStyle{Blue,Thick},PerformanceGoal->"Quality",PlotPoints60],ParametricPlot3D[ζ[t],{t,0,2π},PlotStyle{DarkYellow,Thick}],Graphics3D[{Orange,Ball[

[t1],0.05]}],Graphics3D[{Blue,Ball[

[t1],0.05]}],Graphics3D[{Black,Thick,Line[{

[t1],ζ[t1],

[t1]}]}],Graphics3D[{DarkYellow,Ball[ζ[t1],0.05]}]},AxesTrue,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),BoxedFalse,AxesOrigin{0,0,0},BoxRatiosAutomatic,ImageSize700,PlotRange{{-3,3},{-3,3},{-1,2}},ViewPoint{1.3,-2.4,2.},PlotLabelStyle[Framed["Offset curves"],16,Blue,BackgroundYellow]],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacement->Top

Out[]=

	Positive Offset
	Negative Offset
	Original function
	Generatrix line

Out[]=

Value (t1)