In[]:=
TschirnhausenCurveStudy::usage="Do a complete study of the curve of Tschirnhausen that is on the plane x = y and it is: {3(​
2
t
- 3)​, 3(​
2
t
- 3)​, t(​
2
t
- 3)​}. With respect to the origin. References: Personal."
Out[]=
Do a complete study of the curve of Tschirnhausen that is on the plane x = y and it is: {3(​
2
t
- 3)​, 3(​
2
t
- 3)​, t(​
2
t
- 3)​}. With respect to the origin. References: Personal.
Desarrollo:
Pedal curve:
In[]:=
ζ[t_]:={3(
2
t
-3),3(
2
t
-3),t(
2
t
-3)};​​ζ[t]+({0,0,0}-ζ[t]).
ζ'[t]
Norm[ζ'[t]]
ζ'[t]
Norm[ζ'[t]]
//FullSimplify
Out[]=
-
3(-3+
2
t
)(-
2
Abs[3-3
2
t
]
+6t(23t+
3
t
-24Re[t]))
2
Abs[3-3
2
t
]
+72tConjugate[t]
,-
3(-3+
2
t
)(-
2
Abs[3-3
2
t
]
+6t(23t+
3
t
-24Re[t]))
2
Abs[3-3
2
t
]
+72tConjugate[t]
,-
9t(-3+
2
t
)(-12+11
2
t
+
4
t
-
4
Abs[t]
-8tConjugate[t]+
2
Conjugate[t]
)
2
Abs[3-3
2
t
]
+72tConjugate[t]

In[]:=
-
3(-3+
2
t
)(-
2
Abs[3-3
2
t
]
+6t(23t+
3
t
-24Re[t]))
2
Abs[3-3
2
t
]
+72tConjugate[t]
,-
3(-3+
2
t
)(-
2
Abs[3-3
2
t
]
+6t(23t+
3
t
-24Re[t]))
2
Abs[3-3
2
t
]
+72tConjugate[t]
,-
9t(-3+
2
t
)(-12+11
2
t
+
4
t
-
4
Abs[t]
-8tConjugate[t]+
2
Conjugate[t]
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
;
In[]:=
​​SwatchLegend[{Black,Red,Blue,DarkGreen},{"Tschirnhausen curve","Pedal curve","Tangent line","Orthogonal line to the tangent line"}]​​Manipulateζ[t_]:={3(
2
t
-3),3(
2
t
-3),t(
2
t
-3)};
ζ
11
[t_]:=-
3(-3+
2
t
)(-
2
Abs[3-3
2
t
]
+6t(23t+
3
t
-24Re[t]))
2
Abs[3-3
2
t
]
+72tConjugate[t]
,-
3(-3+
2
t
)(-
2
Abs[3-3
2
t
]
+6t(23t+
3
t
-24Re[t]))
2
Abs[3-3
2
t
]
+72tConjugate[t]
,-
9t(-3+
2
t
)(-12+11
2
t
+
4
t
-
4
Abs[t]
-8tConjugate[t]+
2
Conjugate[t]
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
;ℒ[t_]:=ζ[t]+vζ'[t];​​Show[ParametricPlot3D[ζ[t],{t,-π,π},PlotStyle{Black,Thick}],​​Graphics3D[{Black,Ball[ζ[k1],0.1]}],Graphics3D[{Black,Ball[{0,0,0},0.1]}],​​ParametricPlot3D[
ζ
11
[t],{t,-π,π},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],​​Graphics3D[{Red,Ball[
ζ
11
[k1],0.1]}],​​ParametricPlot3D[ℒ[k1],{v,-π,π},PlotStyle->{Blue,Thick}],Graphics3D[{DarkGreen,Thick,InfiniteLine[{{0,0,0},
ζ
11
[k1]}]}]​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,5.},BoxRatiosAutomatic,PlotRange{{-10,6},{-10,8},{-6,6}},ImageSize900​​],{k1,-π,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop,SaveDefinitions->True
Out[]=
Tschirnhausen curve
Pedal curve
Tangent line
Orthogonal line to the tangent line
Out[]=
​
k1
​
​
​
​
Negative-pedal curve:
In[]:=
ζ[t_]:={3(
2
t
-3),3(
2
t
-3),t(
2
t
-3)};​​{0,0,0}-({0,0,0}-ζ[t]).
ζ'[t]
Norm[ζ'[t]]
ζ'[t]
Norm[ζ'[t]]
//FullSimplify
Out[]=

18
2
t
(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
,
18
2
t
(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
,
9t(-1+
2
t
)(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]

In[]:=

18
2
t
(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
,
18
2
t
(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
,
9t(-1+
2
t
)(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
;
In[]:=
​​SwatchLegend[{Black,Red,Blue,DarkGreen},{"Tschirnhausen curve","Negative-pedal curve","Normal line to the Tschirnhausen curve","Orthogonal line to the normal line"}]​​Manipulateζ[t_]:={3(
2
t
-3),3(
2
t
-3),t(
2
t
-3)};​​
ζ
22
[t_]:=
18
2
t
(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
,
18
2
t
(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
,
9t(-1+
2
t
)(-33+8
2
t
+
4
t
)
2
Abs[3-3
2
t
]
+72tConjugate[t]
;​​Show[ParametricPlot3D[ζ[t],{t,-π,π},PlotStyle{Black,Thick}],​​Graphics3D[{Black,Ball[ζ[k1],0.1]}],Graphics3D[{Black,Ball[{0,0,0},0.1]}],​​ParametricPlot3D[
ζ
22
[t],{t,-π,π},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],​​Graphics3D[{Red,Ball[
ζ
22
[k1],0.1]}],​​Graphics3D[{DarkGreen,Thick,InfiniteLine[{{0,0,0},
ζ
22
[k1]}]}],Graphics3D[{Blue,Thick,InfiniteLine[{ζ[k1],
ζ
22
[k1]}]}]​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,5},BoxRatiosAutomatic,PlotRange{{-10,6},{-10,8},{-6,6}},ImageSize900​​],{k1,-π,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop,SaveDefinitions->True
Out[]=
Tschirnhausen curve
Negative-pedal curve
Normal line to the Tschirnhausen curve
Orthogonal line to the normal line
Out[]=
​
k1
​
​
​
​
The pedal curve of the evolute of the curve of Tschirnhausen it matches the negative-pedal curve of the Tschirnhausen curve. We will see it in the next development and animation:
Parametric equations of the evolute of the curve of Tschirnhausen:
​
​
​
​
Osculating sphere of the Tschirnhausen curve:
​
​
​
​
​
​
Tschirnhausen involute: