InterestingConstructions::usage="(a) Given a curve, γ: {0, u, Sin[u]}, 0 < u < π, calculate (i) its Frenet trihedron, (ii) its curvature, (iii) its torsion. (b) The tubular surface of radius 1 determined by γ, that is, the surface S of vector equation r[u, v] = [u] + Cos[v] [u] + Sin[v] [u], 0 < u < π, 0 ≤ v < 2π, where [u] its position vector of γ. It is requested: 1 Express the surface base {, } in the Frenet trihedron of γ. 2 Calculate the first fundamental form of S. 3 Discuss if any of the lines : S ⋂ y = , : S ⋂ {x = 1}, : S ⋂ {z = 0} it is an asymptotic or geodesic line of S, reasoning the answer. References: www2.caminos.upm.es; Personal. ";
R
N
B
R
g
u
g
v
γ
1
π
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γ
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γ
3
Development (a):
(i) Frenet triherdon:
In[]:=
γ[u_]:={0,u,Sin[u]};
γ'[u]
Norm[γ'[u]]
(γ'[u]γ''[u])γ'[u]
Norm[γ'[u]γ''[u]]Norm[γ'[u]]
γ'[u]γ''[u]
Norm[γ'[u]γ''[u]]
In[]:=
T[u_]:=0,,;
1
1+
2
Abs[Cos[u]]
Cos[u]
1+
2
Abs[Cos[u]]
In[]:=
N1[u_]:=0,,-;
Cos[u]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
In[]:=
B[u_]:=-,0,0;
Sin[u]
Abs[Sin[u]]
(ii) Its curvature:
In[]:=
κ=
Norm[γ'[u]γ''[u]]
3
Norm[γ'[u]]
Abs[Sin[u]]
3/2
(1+)
2
Abs[Cos[u]]
(iii) Its torsion:
In[]:=
χ=
Det[{γ'[u],γ''[u],γ'''[u]}]
2
Norm[γ'[u]γ''[u]]
0;
It’s a flat curve.
Development (b):
1) Express the surface base { , } in the Frenet trihedron of γ (Tube surface: The normal plane subtended by N and B is used. It is a surface that envelops the curve as if it were a cable or hose of radius r1):
g
u
g
v
In[]:=
γ[u_]:={0,u,Sin[u]};r1=1;r[u,v]=γ[u]+r1(Cos[v]N1[u]+Sin[v]B[u])
-,u+,Sin[u]-;
Sin[u]Sin[v]
Abs[Sin[u]]
Cos[u]Cos[v]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
Cos[v]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
In[]:=
ParametricPlot3D-,u+,Sin[u]-,{u,0,π},{v,0,2π}
Sin[u]Sin[v]
Abs[Sin[u]]
Cos[u]Cos[v]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
Cos[v]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
Out[]=
2) Calculate the first fundamental form of S:
In[]:=
ResourceFunction["FirstFundamentalFormCoefficients"]-,u+,Sin[u]-,{u,v}//FullSimplify
Sin[u]Sin[v]
Abs[Sin[u]]
Cos[u]Cos[v]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
Cos[v]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
Out[]=
++,(Cos[v]Sin[u]Sin[v](Abs[Sin[u]](2(1+)Cos[u](-Cos[2u])-Abs[Cos[u]](3+Cos[2u])[Cos[u]])-(1+)(-)Sin[2u][Sin[u]]))2,+
3
(1+)
2
Abs[Cos[u]]
2
Cos[u]
2
Sin[v]
2
(Abs[Sin[u]]-Sin[u][Sin[u]])
′
Abs
2
Cos[u]-Abs[Sin[u]]Cos[v](Cos[u]+Cos[u]+Abs[Cos[u]][Cos[u]])+(1+)Cos[u]Cos[v]Sin[u][Sin[u]]
3/2
(1+)
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Cos[u]]
2
Sin[u]
′
Abs
2
Abs[Cos[u]]
′
Abs
2
+Abs[Sin[u]]Cos[v]((1+)Cos[2u]+Abs[Cos[u]]Cos[u][Cos[u]])-(1+)Cos[v]Sin[u][Sin[u]]
3/2
(1+)
2
Abs[Cos[u]]
2
Abs[Sin[u]]
2
Abs[Cos[u]]
2
Sin[u]
′
Abs
2
Abs[Cos[u]]
2
Cos[u]
′
Abs
3
(1+)
2
Abs[Cos[u]]
4
Abs[Sin[u]]
2
Abs[Cos[u]]
2
Abs[Cos[u]]
2
Sin[u]
′
Abs
2
Abs[Cos[u]]
2
Abs[Cos[u]]
2
Cos[u]
′
Abs
2
(1+)
2
Abs[Cos[u]]
3
Abs[Sin[u]]
2
Sign[Sin[u]]
2
Cos[v]
(1+)
2
Cos[u]
2
Sin[v]
1+Cos[u]Cos[Conjugate[u]]
They depend on the value of u and v.
3)Discussifanyofthelines:Sy=,:S⋂{x=1},:S⋂{z=0}itisanasymptoticorgeodesiclineofS,reasoningtheanswer.:
γ
1
π
2
γ
2
γ
3
Sy=
π
2
In[]:=
Solveu+==,u
Cos[u]CosSin[u]
π
2
1+
Abs[Sin[u]]2
Abs[Cos[u]]
π
2
Out[]=
u
π
2
In[]:=
ParametricPlot3D-,,Sin-,{v,0,2π}
SinSin[v]
π
2
AbsSin
π
2
π
2
π
2
Cos[v]Sin
π
2
1+
AbsSin2
AbsCos
π
2
π
2
Out[]=
In[]:=
ShowParametricPlot3D-,u+,Sin[u]-,{u,0,π},{v,0,2π},ContourPlot3Dy==,{x,-5,5},{y,-5,5},{z,-5,5}
Sin[u]Sin[v]
Abs[Sin[u]]
Cos[u]Cos[v]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
Cos[v]Sin[u]
1+
Abs[Sin[u]]2
Abs[Cos[u]]
π
2
Out[]=
It must be a geodesic curve because it is the shortest and straightest route between two points on the surface, because it is a circumference.
S ⋂ {x = 1}:
It is a sinusoidal form, it must be a geodesic (it is on the surface of the tube since x = 1 and the radius of the tube (surface) is 1).
S ⋂ {z = 0}:
It has the behavior of a semicubic parabola, but is on the surface of tube, must be a geodesic line.
Here the geodesic lines of the surface appear, as you can see from circular shapes and some curves that start from a vertex (origin) in semicubical parabolic shape and curves that sinusoidal shape. In fact, all the curves are part of the geodesic lines from different angles of view. A geodesic line can also be asymptotic if it is a straight line, none of the curves studied above is straight line.