SearchTwoCurves::usage="(a) A curve γ is defined by the parameterization {Cos[t], Sin[t], f[t]}. Calculate the function f[t] so that the main normals of γ are parallel to the XY plane. Calculate the torsion and curvature in that case. (b) Determine the function ϕ[t] so that ϕ[0] = 1. And the normal planes to the curve {Sin[t, Sin[t] Cos[t], ϕ[t]} pass through the origin. Calculate the curvature and torsion in such a case. References: www2.caminos.upm.es; Personal.";
2
]
Development of (a):
Main normal:
In[]:=
γ[t_]:={Cos[t],Sin[t],f[t]};R[t_]:=γ[t]+λ((γ'[t]γ''[t])γ'[t]);
The director vector of the main normal it must be orthogonal to the normal vector of the XY plane:
In[]:=
((γ'[t]γ''[t])γ'[t]).{0,0,1}==0
Out[]=
2
Cos[t]
′′
f
2
Sin[t]
′′
f
In[]:=
DSolve[f''[t]==0,f[t],t]
Out[]=
{{f[t]+t}}
1
2
Calculate the torsion and curvature:
Torsion:
It’s a family of curves:
In[]:=
γ[t_]:={Cos[t],Sin[t],+t};χ=
1
2
Det[{γ'[t],γ''[t],γ'''[t]}]
2
(Norm[γ'[t]γ''[t]])
2
2
Cos[t]
2
2
Sin[t]
2
Abs[Cos[t]]
2
2
Abs[Sin[t]]
2
2
Abs[+]
2
Cos[t]
2
Sin[t]
2
2
2
Curvature:
In[]:=
γ[t_]:={Cos[t],Sin[t],+t};κ=
1
2
Norm[γ'[t]γ''[t]]
3
Norm[γ'[t]]
2
Abs[Cos[t]]
2
2
Abs[Sin[t]]
2
2
Abs[+]
2
Cos[t]
2
Sin[t]
3/2
(++)
2
Abs[]
2
2
Abs[Cos[t]]
2
Abs[Sin[t]]
2
2
3
+1
2
2
1
2
2
Development of (b):
Normal plane:
In[]:=
ζ[t_]:={,Sin[t]Cos[t],ϕ[t]};x1[t_]:=;y1[t_]:=Sin[t]Cos[t];(x-0)x1'[t]+(y-0)y1'[t]+(z-0)ϕ'[t]==0
2
Sin[t]
2
Sin[t]
2xCos[t]Sin[t]+y(-)+z[t]0;
2
Cos[t]
2
Sin[t]
′
ϕ
The plane normal to a curve at a given point is the plane perpendicular to the tangent vector of the curve at that point. That is , the normal vector of the plane is the tangent vector of the curve at that point.
The curve and the plane intersect at a point, then:
2Cos[t]Sin[t]+Sin[t]Cos[t](-)+ϕ[t]ϕ'[t]==0;
2
Sin[t]
2
Cos[t]
2
Sin[t]
Symplifying, it remains:
Cos[t]Sin[t]+ϕ[t]ϕ'[t]==0;
DSolve[{Cos[t]Sin[t]+ϕ[t]ϕ'[t]==0,ϕ[0]==1},ϕ[t],t]
Out[]=
ϕ[t]
2
Cos[t]
ϕ[t]=Cos[t];
Torsion:
ζ[t_]:={,Sin[t]Cos[t],Cos[t]};
2
Sin[t]
The graph of ζ[t] is Viviani’s curve:
ParametricPlot3D[{,Sin[t]Cos[t],Cos[t]},{t,0,2π}]
2
Sin[t]
Out[]=
Torsion:
In[]:=
ζ[t_]:={,Sin[t]Cos[t],Cos[t]};χ=//FullSimplify
2
Sin[t]
Det[{ζ'[t],ζ''[t],ζ'''[t]}]
2
(Norm[ζ'[t]ζ''[t]])
6Sin[t]
4++4
2
Abs[Cos[t](-2+Cos[2t])]
6
Abs[Sin[t]]
6Sin[t]
4++4
2
Cos[t]
2
(-2+Cos[2t])
6
Sin[t]
It depends on the value of t.
Curvature:
In[]:=
ζ[t_]:={,Sin[t]Cos[t],Cos[t]};κ=//FullSimplify
2
Sin[t]
Norm[ζ'[t]ζ''[t]]
3
Norm[ζ'[t]]
4++4
2
Abs[Cos[t](-2+Cos[2t])]
6
Abs[Sin[t]]
3/2
(++)
2
Abs[Cos[2t]]
2
Abs[Sin[t]]
2
Abs[Sin[2t]]
4++4
2
Cos[t]
2
(-2+Cos[2t])
6
Sin[t]
3
2
(1+)
2
Sin[t]
It depends on the value of t.
Animation of one element of family of curves (a):
Manipulate[γ[t_]:={Cos[t],Sin[t],1-3t};R[t_]:=γ[t]+λ((γ'[t]γ''[t])γ'[t]);Show[{ParametricPlot3D[γ[t],{t,-2π,t1},PlotStyle->DarkGreen]/.Line[pts_,rest___]:>Tube[pts,0.025,rest],ParametricPlot3D[R[t1],{λ,-2,2},PlotStyle->Orange]/.Line[pts_,rest___]:>Tube[pts,0.025,rest],Graphics3D[{DarkGreen,Ball[γ[t1],0.08]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-5,5},{-5,5},{-10,10}},ViewPoint{1.3,-2.4,0.5},ImageSize900],Style["Element of family of curves and main normal line:",Bold,Large],{{t1,-2π+0.00002,"Value (t1)"},-2π+0.00001,2π,0.00001},ControlPlacementTop]
Out[]=
Animation (b):