Curvesinsynchronizedmotion::usage="The curve C is given whose equations are x = 3t, y = 3
2
t
​, z = 2
3
t
. 1 Find the equations of the tangent at a point M of the curve, and determine A and B where the tangent cuts to the XOY, XOZ planes. 2 Prove that
MA
MB
it is constant and find the geometric locus of A and B when M describes the curve C. 3 Find the lenght of the arc OM. 4 From M the perpendicular MP to XOY is drawn, and MQ is taken in the direction of M to P, and equal the arc OM. Find the geometric locus of Q.​References: Problemas de Geometría Analítica y Diferencial, Ubaldo Usunáriz Balanzategui, Ignacio Usunáriz Sala, Prob.- J16; Personal";
Development :
In[]:=
ζ[t_]:={3t,3
2
t
,2
3
t
};​​ℒ:{3t,3
2
t
,2
3
t
}+v{3,6t,6
2
t
}=={3t+3v,3
2
t
+6vt,2
3
t
+6
2
t
v};​​Solve[2
3
t
+6
2
t
v==0,v]​​
Out[]=
v-
t
3

In[]:=
A=3t+3-
t
3
,3
2
t
+6-
t
3
t,0
Out[]=
{2t,
2
t
,0}
In[]:=
Solve[3
2
t
+6vt==0,v]
Out[]=
v-
t
2

In[]:=
B=3t+3-
t
2
,0,2
3
t
+6
2
t
-
t
2

Out[]=

3t
2
,0,-
3
t

In[]:=
MA
MB
==
2
(3t-2t)
+
2
(3
2
t
-
2
t
)
+
2
(2
3
t
-0)
2
3t-
3t
2

+
2
(3
2
t
-0)
+
2
(2
3
t
+
3
t
)
In[]:=
Simplify
MA
MB

2
t
+4
4
t
+4
6
t
9
2
t
4
+9
4
t
+9
6
t

MA
MB

2
3
;
Geometric locus when the point A moves (observing the coordinates of the point):
2
x
-4y==0;z==0;
Verification:
2
(2t)
-4(
2
t
)==0;
That is, point A moves on the curve:
x,
2
x
4
,0;
Geometric locus when the point B moves (observing the coordinates of the point):
8
3
x
+27z==0;y==0;
Verification:
8
3
3t
2
+27(-
3
t
)==0;
That is, point B moves on the curve:
x,0,-
8
27
3
x
;
OM arc length:
In[]:=
x[t_]:=3t;y[t_]:=3
2
t
;z[t_]:=2
3
t
;​​s[t]==
t
∫
0
2
(x'[t])
+
2
(y'[t])
+
2
(z'[t])
t
Out[]=
s[t]t(3+2
2
t
) if
2
t
+
4
t
+
2
Conjugate[t]
+
4
Conjugate[t]
≥-
1
2
Also:
s[t]=3t+2
3
t
;
s=x+z;
Geometric locus of Q:
The Q point goes down a distance x + z from M, so its coordinates are Q{x, y, z - (x + z)} = Q{x, y, - x}.
Then the point Q moves on the plane x + z = 0 ,following the trajectory of the original curve.
The curve is {3t, 3
2
t
, 2
3
t
} it is observed that
2
x
-3y=0
.
So the geometric locus of Q is: x + z = 0;
2
x
-3y=0
.
That is, point Q moves on the curve:
x,
2
x
3
,-x;
In[]:=
SwatchLegend[{Blue,LightBlue,Yellow,Green},{"The main curve and its tangent","The plane z = 0 where the geometric locus of point A develops (black curve)","The plane y = 0 where the geometric locus of point B develops (purple curve)","The plane x + z = 0 where the geometric locus of point Q develops (orange curve)"}]​​ManipulateClearAll;ζ[t_]:={3t,3
2
t
,2
3
t
};
ζ
1
[t_]:={2t,
2
t
,0};
ζ
2
[t_]:=
3t
2
,0,-
3
t
;
ζ
3
[t_]:={3t,3
2
t
,-3t};​​​​Show[{Graphics3D[{Blue,Point[ζ[t1]],Arrowheads[.03],Arrow[Tube[{ζ[t1],ζ[t1]+0.1ζ'[t1]}]]}],​​ContourPlot3D[z0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,PerformanceGoal"Quality",ContourStyle->{Opacity[0.2],Blue}],​​ContourPlot3D[y0,{x,-5,5},{y,-5,5},{z,-10,10},ContourStyle->{Opacity[0.2],Yellow},MeshFalse,PerformanceGoal"Quality"],​​ContourPlot3D[x+z0,{x,-20,20},{y,-15,15},{z,-5,5},ContourStyle->{Opacity[0.2],Green},MeshFalse,PerformanceGoal"Quality"],​​ParametricPlot3D[ζ[t1]+tζ'[t1],{t,-2,2},PlotStyle{Blue,Thick}],​​Graphics3D[{Black,Dashed,Thick,Line[{ζ[x],
ζ
3
[x]}]}]/.xt1,​​Graphics3D[{Blue,Ball[ζ[t1],0.3]}],Graphics3D[{Black,Ball[
ζ
1
[t1],0.3]}],Graphics3D[{Purple,Ball[
ζ
2
[t1],0.3]}],Graphics3D[{Orange,Ball[
ζ
3
[t1],0.3]}],​​ParametricPlot3D[ζ[t],{t,-2,t1},PlotStyle{Blue,Thick}],​​ParametricPlot3D[
ζ
1
[t],{t,-2,t1},PlotStyle{Blue,Black}],​​ParametricPlot3D[
ζ
2
[t],{t,-2,t1},PlotStyle{Blue,Purple}],​​ParametricPlot3D[
ζ
3
[t],{t,-2,t1},PlotStyle{Blue,Orange}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange17.7,ViewPoint{-3,3,2.},ImageSize900],Style["The main curve and the geometric loci of points A, B y Q:",Bold,Medium],{{t1,-1.9,"Value (t1)"},-2+0.01,2,0.01},ControlPlacementTop
Out[]=
The main curve and its tangent
The plane z = 0 where the geometric locus of point A develops (black curve)
The plane y = 0 where the geometric locus of point B develops (purple curve)
The plane x + z = 0 where the geometric locus of point Q develops (orange curve)
Out[]=
​
The main curve and the geometric loci of points A, B y Q:
Value (t1)