Studyofadeltoid::usage="Complete study of a deltoid found in the plane x = z : {4 Cos[t] + 2 Cos[2t], 4 Sin[t]- 2 Sin[2t], 4 Cos[t] + 2 Cos[2t] }. With respect to the origin. References: Wolfram MathWorld; Personal"
Development:
Pedal curve:
In[]:=
ζ[t_]:={4Cos[t]+2Cos[2t],4Sin[t]-2Sin[2t],4Cos[t]+2Cos[2t]};​​ζ[t]+({0,0,0}-ζ[t]).
ζ'[t]
Norm[ζ'[t]]
ζ'[t]
Norm[ζ'[t]]
//FullSimplify
(8(4
2
Abs[Cos[t]-Cos[2t]]
(2Cos[t]+Cos[2t])+8
2
Abs[Sin[t]+Sin[2t]]
(2Cos[t]+Cos[2t])-4(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
))(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
),128Cos
t
2
(2
2
Abs[Cos[t]-Cos[2t]]
+4
2
Abs[Sin[t]+Sin[2t]]
+
2
(1+2Cos[t])
(-3+8Cos[t]+Cos[2t]))
3
Sin
t
2

(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
),(8(4
2
Abs[Cos[t]-Cos[2t]]
(2Cos[t]+Cos[2t])+8
2
Abs[Sin[t]+Sin[2t]]
(2Cos[t]+Cos[2t])-4(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
))(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
);​​
ζ
11
[t_]:=(8(4
2
Abs[Cos[t]-Cos[2t]]
(2Cos[t]+Cos[2t])+8
2
Abs[Sin[t]+Sin[2t]]
(2Cos[t]+Cos[2t])-4(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
))(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
),128Cos
t
2
(2
2
Abs[Cos[t]-Cos[2t]]
+4
2
Abs[Sin[t]+Sin[2t]]
+
2
(1+2Cos[t])
(-3+8Cos[t]+Cos[2t]))
3
Sin
t
2

(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
),(8(4
2
Abs[Cos[t]-Cos[2t]]
(2Cos[t]+Cos[2t])+8
2
Abs[Sin[t]+Sin[2t]]
(2Cos[t]+Cos[2t])-4(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
))(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
);
In[]:=
​​SwatchLegend[{Black,Red,Blue,DarkGreen},{"Deltoid","Pedal curve","Tangent line","Orthogonal line to the tangent line"}]​​Manipulateζ[t_]:={4Cos[t]+2Cos[2t],4Sin[t]-2Sin[2t],4Cos[t]+2Cos[2t]};
ζ
11
[t_]:=(8(4
2
Abs[Cos[t]-Cos[2t]]
(2Cos[t]+Cos[2t])+8
2
Abs[Sin[t]+Sin[2t]]
(2Cos[t]+Cos[2t])-4(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
))(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
),128Cos
t
2
(2
2
Abs[Cos[t]-Cos[2t]]
+4
2
Abs[Sin[t]+Sin[2t]]
+
2
(1+2Cos[t])
(-3+8Cos[t]+Cos[2t]))
3
Sin
t
2

(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
),(8(4
2
Abs[Cos[t]-Cos[2t]]
(2Cos[t]+Cos[2t])+8
2
Abs[Sin[t]+Sin[2t]]
(2Cos[t]+Cos[2t])-4(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
))(
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
);ℒ[t_]:=ζ[t]+vζ'[t];​​Show[ParametricPlot3D[ζ[t],{t,0,2π},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{{-8,6},{-8,8},{-6,6}},ImageSize900​​],{k1,-π,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
Deltoid
Pedal curve
Tangent line
Orthogonal line to the tangent line
Out[]=
​
k1
​
​
​
​
Negative-pedal curve:
In[]:=
ζ[t_]:={4Cos[t]+2Cos[2t],4Sin[t]-2Sin[2t],4Cos[t]+2Cos[2t]};​​{0,0,0}-({0,0,0}-ζ[t]).
ζ'[t]
Norm[ζ'[t]]
ζ'[t]
Norm[ζ'[t]]
//FullSimplify

32(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
,-
8Cos
t
2

2
(1+2Cos[t])
(-3+8Cos[t]+Cos[2t])
3
Sin
t
2

2
Abs[Cos[t]-Cos[2t]]
+2
2
Abs[Sin[t]+Sin[2t]]
,
32(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
;​​
ζ
22
[t_]:=
32(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
,-
8Cos
t
2

2
(1+2Cos[t])
(-3+8Cos[t]+Cos[2t])
3
Sin
t
2

2
Abs[Cos[t]-Cos[2t]]
+2
2
Abs[Sin[t]+Sin[2t]]
,
32(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
;
In[]:=
​​SwatchLegend[{Black,Red,Blue,DarkGreen},{"Deltoid","Negative-pedal curve","Normal line of the deltoid","Orthogonal line to the normal line"}]​​Manipulateζ[t_]:={4Cos[t]+2Cos[2t],4Sin[t]-2Sin[2t],4Cos[t]+2Cos[2t]};​​
ζ
22
[t_]:=
32(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
,-
8Cos
t
2

2
(1+2Cos[t])
(-3+8Cos[t]+Cos[2t])
3
Sin
t
2

2
Abs[Cos[t]-Cos[2t]]
+2
2
Abs[Sin[t]+Sin[2t]]
,
32(-3+8Cos[t]+Cos[2t])
2
(Sin[t]+Sin[2t])
2
Abs[4Cos[t]-4Cos[2t]]
+32
2
Abs[Sin[t]+Sin[2t]]
;​​Show[ParametricPlot3D[ζ[t],{t,0,2π},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{{-8,6},{-8,8},{-6,6}},ImageSize900​​],{k1,-π,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop
​
​
​
​
The pedal curve of the evolute of the deltoid coincides with the negative-pedal curve of the deltoid. We will see it in the following development and animation:
Parametric equations of the deltoid evolute:
​
​
​
​
Osculating circumference of the deltoid:
The center of the osculating circumference is the center of curvature. The tangent line to the evolute is normal to the deltoid, that is, the evolute is the envelope of the normals of the deltoid. That is, the evolute (the major deltoid in green) is tangent to each element of the family of lines at some point, and these tangent points together form the complete envelope.
​
​
​
​
​
​
Involute of the deltoid:
It is observed how the tangent lines to the deltoid are always normal to the involute.