EnvelopeNormalPlanes::usage="Find the envelope, the characteristics, and the recoil edge of the family of normal planes to a given curve. References: Mishchenko, A.S., Solovyev, Yu.P., Fomenko, A.T.Problems in Differential Geometry and Topology. Mir Publishers Moscow: p.40, 1985; Personal.";
The curve ζ[t] = {5 t,
2
t
,
3
t
}. will be used.
Family of normal planes:
In[]:=
Q[t_]:={5t,
2
t
,
3
t
};​​x[t_]:=5t;​​y[t_]:=
2
t
;​​z[t_]:=
3
t
;​​(x-x[t])x'[t]+(y-y[t])y'[t]+(z-z[t])z'[t]==0​​
In[]:=
Expand[5(-5t+x)+2t(-
2
t
+y)+3
2
t
(-
3
t
+z)0]
Out[]=
-25t-2
3
t
-3
5
t
+5x+2ty+3
2
t
z0
Characteristics:
In[]:=
Solve[-25t-2
3
t
-3
5
t
+5x+2ty+3
2
t
z0,
∂
t
(-25t-2
3
t
-3
5
t
+5x+2ty+3
2
t
z)==0,{x,y}]
Out[]=
x
1
5
(-4
3
t
-12
5
t
+3
2
t
z),y
1
2
(25+6
2
t
+15
4
t
-6tz)

1
5
(-4
3
t
-12
5
t
+3
2
t
z),
1
2
(25+6
2
t
+15
4
t
-6tz),z;
Envelope :
In[]:=
Eliminatex==
1
5
(-4
3
t
-12
5
t
+3
2
t
z),y==
1
2
(25+6
2
t
+15
4
t
-6tz),z==z,t
In[]:=
Simplify[-21904000000-549086400
2
x
-52734375
4
x
+8808960000y+88920000
2
x
y-1417036800
2
y
-3600000
2
x
2
y
+113973248
3
y
-4583424
4
y
+73728
5
y
+3312432000xz+25312500
3
x
z-799954560xyz+64396800x
2
y
z-1728000x
3
y
z-121320000
2
z
-116133750
2
x
2
z
+29131200y
2
z
+9112500
2
x
y
2
z
-2331648
2
y
2
z
+62208
3
y
2
z
+12740220x
3
z
-1020600xy
3
z
+4100625
4
z
-656100y
4
z
+26244
2
y
4
z
-393660x
5
z
0]
25312500
3
x
z+
2
(25-2y)
(-35046400-685056
2
y
+18432
3
y
-194112
2
z
+6561
4
z
+64y(132608+243
2
z
))45x(1171875
3
x
+10x(1220192+8000
2
y
+258075
2
z
-50y(3952+405
2
z
))+4z(-1842400-357760
2
y
+9600
3
y
-70779
2
z
+2187
4
z
+y(4444192+5670
2
z
)));
The cartesian equation is very complicated, we will use:

1
5
(-4
3
t
-12
5
t
+3
2
t
z),
1
2
(25+6
2
t
+15
4
t
-6tz),z;
Recoil edge:
In[]:=
Solve-25t-2
3
t
-3
5
t
+5x+2ty+3
2
t
z0,
∂
t
(-25t-2
3
t
-3
5
t
+5x+2ty+3
2
t
z)==0,
∂
t,t
(-25t-2
3
t
-3
5
t
+5x+2ty+3
2
t
z)==0,{x,y,z}
Out[]=
x
2
5
(
3
t
+9
5
t
),y
1
2
(25-6
2
t
-45
4
t
),z2(t+5
3
t
)

2
5
(
3
t
+9
5
t
),
1
2
(25-6
2
t
-45
4
t
),2(t+5
3
t
);
Animator:
In[]:=
​​SwatchLegend[{Yellow,Blue,Green,Red},{"Envelope of the normal planes","Characteristics","Recoil edge","Normal planes"}]​​ManipulateClearAll;​​ShowContourPlot3D[-25t1-2
3
t1
-3
5
t1
+5x+2t1y+3
2
t1
z0,{x,-10,10},{y,-20,20},{z,-10,10},ContourStyleDirective[Red,Opacity[0.7],Specularity[White,30]],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality"],​​ParametricPlot3D
1
5
(-4
3
t1
-12
5
t1
+3
2
t1
z),
1
2
(25+6
2
t1
+15
4
t1
-6t1z),z,{z,-5,5},PlotStyle{Blue,Thick},​​ParametricPlot3D
1
5
(-4
3
t
-12
5
t
+3
2
t
z),
1
2
(25+6
2
t
+15
4
t
-6tz),z,{t,-1,1},{z,-5,10},BoundaryStyleDirective[Black,Thick],PlotStyleDirective[Yellow,Opacity[0.6],Specularity[White,30]],MeshFalse,PerformanceGoal"Quality",ParametricPlot3D
2
5
(
3
t
+9
5
t
),
1
2
(25-6
2
t
-45
4
t
),2(t+5
3
t
),{t,-0.7,0.7t1},PlotStyle->{Green,Thickness[0.005]}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,7,-2.},ImageSize900,Style["Envelope of normal planes:",Bold,Large],{{t1,-0.9,"Value (t1)"},-1+0.00001,1,0.00001},ControlPlacementTop
Out[]=
Envelope of the normal planes
Characteristics
Recoil edge
Normal planes
Out[]=
​
Envelope of normal planes:
Value (t1)