ConchoidSurface::usage="The definition of a conchoid surface is:​
r
c
​[u, v] = r[u, v] ± k
r[u,v]-P
r[u,v]-P
​, P is the pole, k distance and r[u, v] the base surface.In this notebook we will obtain the conchoid surface of the paraboloid: ​{u Cos[v]​, u Sin[v]​,
2
u
​} with the distance k = 1.5 and as pole P = {0, 0, 1}.Reference: Peternell, M. et al.Conchoid surfaces of rational ruled surfaces. Vienna University of Technology; Personal. ";
Graph:
In[]:=
SwatchLegend[{Orange,Blue,Yellow},{"Positive conchoid surface","Negative conchoid surface","Base surface"}]​​Showr[u_,v_]:={uCos[v],uSin[v],
2
u
};k=1.5;P={0,0,1};r1[u_,v_]:=r[u,v]+k
r[u,v]-P
Norm[r[u,v]-P]
;​​r2[u_,v_]:=r[u,v]-k
r[u,v]-P
Norm[r[u,v]-P]
;{ParametricPlot3D[r1[u,v],{u,0,2π},{v,0,2π},PerformanceGoal"Quality",PlotStyle{Opacity[0.5],Orange},PlotPoints60],​​ParametricPlot3D[r2[u,v],{u,0,2π},{v,0,2π},PerformanceGoal"Quality",PlotStyle{Opacity[0.9],Blue},PlotPoints60],ParametricPlot3D[r[u,v],{u,0,π},{v,0,2π},MeshFalse,PlotStyle->{Yellow,Opacity[0.4]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotLabelStyle[Framed["Conchoid surface of a paraboloid"],16,Blue,BackgroundYellow],PlotRange{{-5,5},{-5,5},{-2,5}},ViewPoint{-2,-2.4,2},ImageSize900
Out[]=
Positive conchoid surface
Negative conchoid surface
Base surface
Out[]=
Explanation:
Cutting the surface:
z=
2
x
+
2
y
;
with the plane y = 0, results:
z=
2
x
;
In[]:=
SwatchLegend[{Orange,Blue,DarkYellow,Red,Black},{"Positive Conchoid","Negative Conchoid","Original function","Pole","Generatrix line"}]​​Manipulatef[t_]:=t;g[t_]:=0;​​h[t_]:=
2
t
;
x
0
=0;
y
0
=0;
z
0
=1;k=1.5;
ζ
+
[t_]:=f[t]+
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,g[t]+
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,;​​
ζ
-
[t_]:=f[t]-
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
,g[t]-
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
;​​
ζ
+
[t_]:=f[t]+
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
+
2
(h[t]-
z
0
)
,g[t]+
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
+
2
(h[t]-
z
0
)
,h[t]+
k(h[t]-
z
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
+
2
(h[t]-
z
0
)
;​​
ζ
-
[t_]:=f[t]-
k(f[t]-
x
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
+
2
(h[t]-
z
0
)
,g[t]-
k(g[t]-
y
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
+
2
(h[t]-
z
0
)
,h[t]-
k(h[t]-
z
0
)
2
(f[t]-
x
0
)
+
2
(g[t]-
y
0
)
+
2
(h[t]-
z
0
)
;​​Show[{ParametricPlot3D[
ζ
+
[t],{t,-2,t1},PlotStyle{Orange,Thick},PerformanceGoal->"Quality"],ParametricPlot3D[
ζ
-
[t],{t,-2,t1},PlotStyle{Blue,Thick},PerformanceGoal->"Quality"],ParametricPlot3D[{f[t],g[t],h[t]},{t,-2,2},PlotStyle{DarkYellow,Thick}],Graphics3D[{Orange,Ball[
ζ
+
[t1],0.05]}],Graphics3D[{Blue,Ball[
ζ
-
[t1],0.05]}],Graphics3D[{Black,Thick,Line[{
ζ
-
[t1],{
x
0
,
y
0
,
z
0
},
ζ
+
[t1]}]}],Graphics3D[{Red,Ball[{
x
0
,
y
0
,
z
0
},0.05]}],Graphics3D[{DarkYellow,Ball[{f[t1],g[t1],h[t1]},0.05]}]},AxesTrue,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),BoxedFalse,AxesOrigin{0,0,0},BoxRatiosAutomatic,ImageSize900,PlotRange{{-4,4},{-1,1},{-2,6}},ViewPoint{1.3,-2.4,1.},PlotLabelStyle[Framed["Conchoid"],16,Blue,BackgroundYellow]],{{t1,-2+0.00002,"Value (t1)"},-2+0.00001,2,0.00001},ControlPlacement->Top​​
Out[]=
Positive Conchoid
Negative Conchoid
Original function
Pole
Generatrix line
Out[]=
​
Value (t1)