In[]:=
TNBNegativePedalCurves::usage="
References:
​Personal.";
All the curves we will write according to t:
T negative-pedal curve of the original:
In[]:=
α[t_]:={Cosh[t],Sinh[t],t};​​ResourceFunction["TangentVector"][α[t],t]//FullSimplify
Out[]=

Sinh[t]
1+Cosh[2t]
,
Cosh[t]
1+Cosh[2t]
,
1
1+Cosh[2t]

In[]:=
​​T[t_]:=
Sinh[t]
1+Cosh[2t]
,
Cosh[t]
1+Cosh[2t]
,
1
1+Cosh[2t]
;​​{0,0,0}-(-α[t].T[t])T[t]//FullSimplify
Out[]=

1
2
Sech[t](t+Sinh[2t])Tanh[t],
1
2
tSech[t]+Sinh[t],
1
2
t
2
Sech[t]
+Tanh[t]
In[]:=
σ
CPT
[t_]:=
1
2
Sech[t](t+Sinh[2t])Tanh[t],
1
2
tSech[t]+Sinh[t],
1
2
t
2
Sech[t]
+Tanh[t];
N negative-pedal curve of the original:
In[]:=
α[t_]:={Cosh[t],Sinh[t],t};​​ResourceFunction["NormalVector"][α[t],t]//FullSimplify
Out[]=
{Sech[t],0,-Tanh[t]}
In[]:=
N1[t_]:={Sech[t],0,-Tanh[t]};​​{0,0,0}-(-α[t].N1[t])N1[t]//FullSimplify
Out[]=
{Sech[t](1-tTanh[t]),0,Tanh[t](-1+tTanh[t])}
In[]:=
σ
CPN
[t_]:={Sech[t](1-tTanh[t]),0,Tanh[t](-1+tTanh[t])};
B negative-pedal curve of the original:
In[]:=
α[t_]:={Cosh[t],Sinh[t],t};​​ResourceFunction["BinormalVector"][α[t],t]//FullSimplify
Out[]=
-
Sinh[t]
1+Cosh[2t]
,
Cosh[t]
1+Cosh[2t]
,-
1
1+Cosh[2t]

In[]:=
​​B[t_]:=-
Sinh[t]
1+Cosh[2t]
,
Cosh[t]
1+Cosh[2t]
,-
1
1+Cosh[2t]
;​​{0,0,0}-(-α[t].B[t])B[t]//FullSimplify
Out[]=

1
2
tSech[t]Tanh[t],-
1
2
tSech[t],
1
2
t
2
Sech[t]

In[]:=
α
CPB
[t_]:=
1
2
tSech[t]Tanh[t],-
1
2
tSech[t],
1
2
t
2
Sech[t]
;
Animations:
T negative-pedal curve:
In[]:=
​​SwatchLegend[{Black,Red,Blue,DarkGreen},{"Original curve","T-negative-pedal curve","Normal line","Othogonal line to the normal line"}]​​Manipulateζ[t_]:={Cosh[t],Sinh[t],t};
ζ
11
[t_]:=
1
2
Sech[t](t+Sinh[2t])Tanh[t],
1
2
tSech[t]+Sinh[t],
1
2
t
2
Sech[t]
+Tanh[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]}],​​Graphics3D[{Blue,Thick,InfiniteLine[{ζ[k1],
ζ
11
[k1]}]}],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,-2.4,2.},BoxRatiosAutomatic,PlotRange{{-8,8},{-8,8},{-3,3}},ImageSize600​​],{k1,-π,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
Original curve
T-negative-pedal curve
Normal line
Othogonal line to the normal line
Out[]=
​
k1
Note: The T negative-pedal curve of the original curve is N pedal curve.
N negative-pedal curve:
In[]:=
SwatchLegend[{Black,Red,Blue,DarkGreen},{"Original curve","N negative-pedal curve","Tangent line","Orthogonal line tothe tangent line"}]​​Manipulate[ζ[t_]:={Cosh[t],Sinh[t],t};
ζ
11
[t_]:={Sech[t](1-tTanh[t]),0,Tanh[t](-1+tTanh[t])};​​Show[ParametricPlot3D[ζ[t],{t,-2π,2π},PlotStyle{Black,Thick}],​​Graphics3D[{Black,Ball[ζ[k1],0.1]}],Graphics3D[{Black,Ball[{0,0,0},0.1]}],​​ParametricPlot3D[
ζ
11
[t],{t,-2π,2π},PlotStyle{Red,Thick},PerformanceGoal->"Quality"],​​Graphics3D[{Red,Ball[
ζ
11
[k1],0.1]}],​​Graphics3D[{Blue,Thick,InfiniteLine[{ζ[k1],
ζ
11
[k1]}]}],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,-2.4,2.},BoxRatiosAutomatic,PlotRange{{-8,8},{-8,8},{-3,3}},ImageSize600​​],{k1,-2π,2π,0.00001},SaveDefinitionsTrue,ControlPlacementTop]
Note: The N negative-pedal curve of the original curve is T pedal curve.
B negative-pedal curve:
Note: This curve presents a mixed behavior of T and N. That’s why no type of line is indicated.