otto[t_] := {Sin[t], Sin[t] Cos[t]}

ParametricPlot[otto[t], {t, 0, 2  Pi}]

cerchio[t_] := {2 Cos[t], 2 Sin[t]}

ParametricPlot[cerchio[t], {t, 0, 2  Pi},  
      AspectRatio  ->  Automatic]

cycloid[t_] := {t - Sin[t], 1 - Cos[t]}

spirale[t_] := {t Cos[t], t Sin[t]}

spirlog[t_] := {E^(t/10)  Cos[t], 
        E^(t/10) Sin[t]}

lemniscata[t_] := {Cos[t]/(1 + Sin[t]^2), 
        Sin[t] Cos[t]/(1 + Sin[t]^2)}

catenaria[t_] := {Cosh[t], t}

trattrice[t_] := {Sin[t], Cos[t] + Log[Tan[t/2]]}

LIBRO di ALFRED GRAY
\(Differential\ Geometry\ of\ Curves\ and\ Surfaces ... 
\)
in biblioteca

ParametricPlot[cycloid[t], {t, -3 Pi, 3  Pi},  
      AspectRatio  ->  Automatic]

ParametricPlot[spirale[t], {t, 0, 5  Pi},  
      AspectRatio  ->  Automatic]

ParametricPlot[spirlog[t], {t, 0, 5  Pi},  
      AspectRatio  ->  Automatic]

ParametricPlot[spirlog[t], {t, 0, 5  Pi},  
      AspectRatio  ->  Automatic]

ParametricPlot[lemniscata[t], {t, 0, 2  Pi},  
      AspectRatio  ->  Automatic]

ParametricPlot[catenaria[t], {t, -5, 5}]

ParametricPlot[{trattrice[t], cerchio[t]}, {t, -2, 2}, 
      AspectRatio -> Automatic]

Plot[{Sin[t], t^2}, {t, -5, 5}, 
      AspectRatio -> Automatic]

D[D[cerchio[t], t], t]

arclength[alpha_][t_] := 
      Sqrt[Simplify[D[alpha[s], s] . D[alpha[s], s]]] /. 
        s -> t

arclength[cerchio][t]

arclength[spirale][t]

Integrate[arclength[spirale][t], {t, 0, 2  Pi}]

NIntegrate[arclength[spirale][t], {t, 0, 2  Pi}]

length[a_, b_][alpha_] := 
      Integrate[arclength[alpha][u], {u, a, b}]

length[0, 2  Pi][otto]

nlength[a_, b_][alpha_] := 
      NIntegrate[arclength[alpha][u], {u, a, b}]

nlength[0, 2  Pi][otto]

length[0, 2  Pi][lemniscata]

arclength[lemniscata][t]

nlength[0, 2  Pi][lemniscata]

Assuming[t > 0, length[1, t][trattrice]]