Wolfram Cloud Document

SurfaceNormalStraightLines::usage=" Find the surface formed by the normals to the surface y = x Tan[x] traced at the points of the line y = x, z =

π

4

.References:A.V.Pogorélov.GeometríaDiferencial.Moscú:EditorialMir,p.127,1977;Personal";

Development:

ManipulateClearAll;r[u_,v_]:={u,uTan[v],v};

r

1

[u_,v_]=D[r[u,v],u];

r

2

[u_,v_]=D[r[u,v],v];n1[u_,v_]:=

r

1

[u,v]

r

2

[u,v]

Norm[

r

1

[u,v]

r

2

[u,v]]

;ζ[u_]:=u,u,

π

4

;ShowParametricPlot3D[ζ[u],{u,-5,t1},PlotStyle{Blue,Thick}],ParametricPlot3Dr[u,v],{u,-5,t1},v,-1.5,1.5

t1

5

,PlotStyle->{Opacity[0.8],Yellow},PerformanceGoal"Quality",Mesh->None,Graphics3D[{Orange,Ball[ζ[u],0.35]}]/.ut1,TableGraphics3DGreen,Point[ζ[u]],Arrowheads[.05],ArrowTubeζ[u],ζ[u]+10n1u,

u

6.6666

,{u,-10,t1,0.3},TableGraphics3DGreen,Point[ζ[u]],Arrowheads[.05],ArrowTubeζ[u],ζ[u]-10n1u,

u

6.6666

,{u,-10,t1,0.3},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange9,ViewPoint{4,3,5},ImageSize700,Style["Surface formed by normal straight lines:",Bold,Large],{{t1,-9.9,"Valor (t1)"},-10+0.01,10,0.01},ControlPlacementTop

Out[]=

Superficie formada por las normales:

Valor (t1)

The normal lines in green, form a hyperbolic paraboloid. The original surface in yellow.