EnvelopeOsculatingPlanes::usage="References: A.S. Mishchenko, Yu. P. Solovyev, A.T. Fomenko, Problems in Differential Geometry and Topology, A.S. Mishchenko, Yu. P. Solovyev, A.T. Fomenko, Mir Publishers: Moscow, p.40, 1985; Personal";
Be the curve Q[t] = {5 t, , }:
2
t
3
t
Family of osculating planes:
In[]:=
Q[t_]:={5t,,};x[t_]:=5t;y[t_]:=;z[t_]:=;Det[{{x-x[t],y-y[t],z-z[t]},{x'[t],y'[t],z'[t]},{x''[t],y''[t],z''[t]}}]==0
2
t
3
t
2
t
3
t
Out[]=
-10+6x-30ty+10z0
3
t
2
t
Another way to do it:
In[]:=
B=Q'[t]Q''[t];({x,y,z}-Q[t]).B==0
In[]:=
Expand[6(-5t+x)-30t(-+y)+10(-+z)0]
2
t
2
t
3
t
Out[]=
-10+6x-30ty+10z0
3
t
2
t
Characteristic:
In[]:=
Solve[-10+6x-30ty+10z0,(-10+6x-30ty+10z)==0,{x,y}]
3
t
2
t
∂
t
3
t
2
t
Out[]=
x,y-
5(2+z)
3
t
3
2
t
--2z
3
t
3t
,-,z;
5(2+z)
3
t
3
2
t
--2z
3
t
3t
Envelope:
In[]:=
Eliminatex==,y==-,z==z,t
5(2+z)
3
t
3
2
t
--2z
3
t
3t
x(4-150y)z+125(15-500y);
2
x
2
z
2
x
2
y
Recoil edge:
In[]:=
Solve-10+6x-30ty+10z0,(-10+6x-30ty+10z)==0,(-10+6x-30ty+10z)==0,{x,y,z}
3
t
2
t
∂
t
3
t
2
t
∂
t,t
3
t
2
t
Out[]=
{{x5t,y,z}}
2
t
3
t
{x,y,z}={5t,,};
2
t
3
t
{5t,,};
2
t
3
t
It is the original curve.
ManipulateClearAll;ShowContourPlot3D[-10+6x-30t1y+10z0,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyleDirective[Red,Opacity[0.7],Specularity[White,30]],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality"],ParametricPlot3D,-,z,{z,-2,2},PlotStyle{Blue,Thick},ContourPlot3D[x(4-150y)z+125(15-500y),{x,-5,5},{y,-5,5},{z,-5,5},BoundaryStyleDirective[Black,Thick],ContourStyleDirective[Yellow,Opacity[0.6],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality"],ParametricPlot3D[{5t,,},{t,-1,t1},PlotStyle->{Green,Thickness[0.005]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange6,ViewPoint{-6,3,5},ImageSize900,Style["Envelope of osculating planes:",Bold,Large],{{t1,-0.9,"Value (t1)"},-1+0.00001,1,0.00001},ControlPlacementTop
3
t1
2
t1
5(2+z)
3
t1
3
2
t1
--2z
3
t1
3t1
2
x
2
z
2
x
2
y
2
t
3
t
Out[]=
The envelope in yellow, the family of osculating planes in red, the characteristic in blue, the recoil edge in green, in this case it is the original curve.
It is observed that when the plane (family of osculating planes) is moved, the charcateristic also moves to make contact with the envelope. The plane and the characteristic follow the route of the recoil edge.
It is observed that when the plane (family of osculating planes) is moved, the charcateristic also moves to make contact with the envelope. The plane and the characteristic follow the route of the recoil edge.