TangentNormalPlaneViviani::usage="Write the equations of the tangent line and the normal plane of the Viviani curve.References: Fedenko, A.S.Problemas de Geometría Diferencial. Editorial Mir, Moscú: p.62, 1981.";
Development:
r
1
[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],
25
2
-
25
2
Cos[t]
;​​
r
2
[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],-
25
2
-
25
2
Cos[t]
;​​
Be t =
π
3
in the upper curve, remains the point P :
In[]:=
r
1
[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],
25
2
-
25
2
Cos[t]
;​​P=
r
1

π
3

Out[]=

15
4
,
5
3
4
,
5
2

Tangent line:
In[]:=
r
1
[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],
25
2
-
25
2
Cos[t]
;​​ℒ:
r
1

π
3
+λ
r
1
'
π
3

Out[]=
ℒ:
15
4
-
5
3
λ
4
,
5
3
4
+
5λ
4
,
5
2
+
5
3
λ
4

Normal plane:
In[]:=
r
1
[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],
25
2
-
25
2
Cos[t]
;​​{x,y,z}-
r
1

π
3
.
r
1
'
π
3
==0
In[]:=
Simplify-
5
4
3
-
15
4
+x+
5
4
-
5
3
4
+y+
5
4
3
-
5
2
+z0
3
xy+
3
z;
In[]:=
​​ManipulateClearAll;
r
1
[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],
25
2
-
25
2
Cos[t]
;​​
r
2
[t_]:=
5
2
+
5
2
Cos[t],
5
2
Sin[t],-
25
2
-
25
2
Cos[t]
;ℒ[t_]:=
15
4
-
5
3
λ
4
,
5
3
4
+
5λ
4
,
5
2
+
5
3
λ
4
;​​ShowContourPlot3D[
3
xy+
3
z,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyle->{Opacity[0.1],Blue},MeshFalse,PerformanceGoal"Quality"],​​ParametricPlot3D[
r
1
[t],{t,0,t1},PlotStyle{Thick,Blue}],​​ParametricPlot3D[
r
2
[t],{t,0,t1},PlotStyle{Thick,Blue}],​​ParametricPlot3D[ℒ[λ],{λ,-10,10t1/(2π)},PlotStyle{Thick,Green}],Graphics3DBlack,Ball
15
4
,
5
3
4
,
5
2
,0.15,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange5,ViewPoint{-3,-2,-1},ImageSize500,Style["Tangent and plane normal to Viviani's curve:",Bold,Large],{{t1,0.1,"Value (t1)"},0.01,2π,0.01},ControlPlacementTop
Out[]=
​
Tangent and plane normal to Viviani's curve:
Value (t1)