CircumferenceInSpace::usage="The curve γ defined by the equations: Cos[u], 1 - Sin[u], - Cos[u] is considered. Prove that it is a circumference. Find the equation of the plane in which it is located, as well as the center and the radius.References: www2.caminos.upm.es; Personal."
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Development:
Equation of the plane that contains it:
Let’s get 3 points:
For u = 0:
,1,-;
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For u = :
π
2
{0,0,0};
For u = :
π
-,1,
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Parametric equations of the plane ρ:
ρ={0,0,0}+s,1,-+t-,1,;
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Cartesian equation of the plane:
In[]:=
Eliminatex==s-t,y==s+t,z==-s+t,{s,t}
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-4z3x;3x+4z==0;
The center:
We take two opposite points like those that correspond to u = 0 and u = π and calculate the midpoint:
C={0,1,0};
The radius:
In[]:=
r=Norm{0,1,0}--,1,
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r=1;
Verification that it is a circumference:
The geometric locus of a circumference is defined as the point that when moving always maintains a constant distance to a fixed point:
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r=Norm{0,1,0}-Cos[u],1-Sin[u],-Cos[u]
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Abs[Cos[u]]
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Abs[Sin[u]]
This proves it.
Animation:
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Manipulateγ[u_]:=Cos[u],1-Sin[u],-Cos[u];Show[{ParametricPlot3D[γ[u],{u,0,u1},PlotStyle->Yellow]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],Graphics3D[{Yellow,Ball[γ[u1],0.035]}],ContourPlot3D[3x+4z==0,{x,-1.5,1.5},{y,-1.5,3},{z,-1.5,1.5},ContourStyle->{Opacity[0.2],Gray},MeshFalse]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-1.5,1.5},{-1.5,3},{-1.5,1.5}},ViewPoint{1.3,-2.4,2.},ImageSize700],Style["Circumference in space:",Bold,Large],{{u1,0.00002,"Value (u1)"},0.00001,2π,0.00001},ControlPlacementTop
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