WorldOfCones::usage="Generalized cone:​

Χ
​(u, v) = Ρ + v

y
​(u)​Wherer Ρ is a fixed point (vertex)​,

y
​(u)​, the directrix curve.The straight line generatrix go through the vertex and the directrix.Referencia: Wolfram MathWorld: Generalized Cone. ";
Straight circular cone:
In[]:=
​​Manipulate[Show[{ParametricPlot3D[{0,0,0}+{vCos[u],vSin[u],v},{u,0,t},{v,-5,5},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow"]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll],Style["Straight circular cone:",Bold,Large],{{t,1,"Value (t)"},0.01,2π,0.01},ControlPlacementTop]
Out[]=
​
Straight circular cone:
Value (t)
Straight elliptic cone:
In[]:=
​​ManipulateShowParametricPlot3D{0,0,0}+
1
2
vCos[u],
1
3
vSin[u],v,{u,0,t},{v,-5,5},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,Style["Straight elliptic cone:",Bold,Medium],{{t,1,"Value (t)"},0.01,2π,0.01},ControlPlacementTop
Out[]=
​
Straight elliptic cone:
Value (t)
Generalized cone 1 (cardioid directrix):
In[]:=
​​ManipulateShowParametricPlot3D{0,0,0}+
1
2
v(1+Cos[u])Cos[u],
1
2
v(1+Cos[u])Sin[u],v,{u,0,t},{v,-5,5},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,Style["Generalized cone 1 (cardioid directrix):",Bold,Large],{{t,1,"Value (t)"},0.01,2π,0.01},ControlPlacementTop
Out[]=
​
Generalized cone 1 (cardioid directrix):
Value (t)
Generalized cone 2 (Lemniscata of Bernouille directrix):
In[]:=
​​ManipulateShowParametricPlot3D{0,0,0}+
1
2
v
Cos[2u]
Cos[u],
1
2
v
Cos[2u]
Sin[u],v,u,-
π
4
,t1,{v,-5,5},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",ParametricPlot3D{0,0,0}+
1
2
v
Cos[2u]
Cos[u],
1
2
v
Cos[2u]
Sin[u],v,u,
3π
4
,t2,{v,-5,5},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,Style["Generalized cone 2 (Lemniscata of Bernouille directrix) :",Bold,Large],{t1,1,"Value (t1)"},-
π
4
+0.01,
π
4
,0.01,Delimiter,{t2,1,"Value (t2)"},
3π
4
+0.01,
5π
4
,0.01,ControlPlacementTop
Out[]=
​
Generalized cone 2 (Lemniscata of Bernouille directrix) :
Value (t1)
Value (t2)
Generalized cone 3 (Spiral of Archimedes directrix):
Generalized cone 4 (Astroid directrix):
Generalized cone 5 (Rose of 8 petals directrix):
Generalized cone 6 (Snail of Pascal directrix):