InvoluteOfCircumference::usage="
​​References:https://e.math.cornell.edu>people;math.ucm.es;​Personal.";
Development:
Arc Length and involute of circumference:
In[]:=
ζ[t_]:={Cos[t],Sin[t]};​​x[t_]:=Cos[t];y[t_]:=Sin[t];​​s[t_]:=
t
∫
0
2
x'[t]
+
2
y'[t]
t;​​ζ[t]-s[t]ζ'[t]​​
ζ
ev
[t_]:={Cos[t]+tSin[t],-tCos[t]+Sin[t]};
Animation:
In[]:=
​​Manipulate[ζ[t_]:={Cos[t],Sin[t]};
ζ
ev
[t_]:={Cos[t]+tSin[t],-tCos[t]+Sin[t]};​​Show[{ParametricPlot[ζ[t],{t,0,2π},PlotStyle{Black,Thick}],ParametricPlot[
ζ
ev
[t],{t,0,t1},PlotStyle->{Blue,Thick}],Graphics[{Blue,PointSize[0.01],Point[
ζ
ev
[t1]]}],Graphics[{Black,Thick,Line[{ζ[t1],
ζ
ev
[t1]}]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y"}),AxesOrigin{0,0},AxesTrue,AspectRatioAutomatic,PlotRange{{-13,8},{-13,10}},ImageSize900],Style["Involute of a circumference:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,4π,0.00001},ControlPlacementTop]
Out[]=
​
Involute of a circumference:
Value (t1)