<< Graphics`Animation`

Parametrisation of a triangle (with almost constant speed)

x[t_] := 
      Which[t < 2, t, t < 3, 4 - t, t ≤ 4, 4 - t]
y[t_] := 
      Which[t < 2, 0, t < 3, t - 2, t ≤ 4, 4 - t]
ShowAnimation[
        Table[Graphics[{AbsolutePointSize[15], RGBColor[1, 1, 0], 
              Point[{x[(m/8)], y[(m/8)]}]}, AspectRatio -> Automatic, 
            Background -> RGBColor[0, 0, 0], 
            PlotRange -> {{- .1, 2.1}, {- .1, 1.1}}], {m, 0, 
            32}]];

Parametrisation of a triangle (with significantly different speeds)

x[t_] := 
      Which[t < 2, t, t < 22, ((42 - t)/20), t ≤ 22.5, 
        45 - 2  t]
y[t_] := 
      Which[t < 2, 0, t < 22, ((t - 2)/20), t ≤ 22.5, 
        45 - 2  t]
ShowAnimation[
        Table[Graphics[{AbsolutePointSize[15], RGBColor[1, 1, 0], 
              Point[{x[(m/4)], y[(m/4)]}]}, AspectRatio -> Automatic, 
            Background -> RGBColor[0, 0, 0], 
            PlotRange -> {{- .1, 2.1}, {- .1, 1.1}}], {m, 0, 
            90}]];

Parametrisation of an ellipse (with variable angular speed)

ω  =  1;
a  =  3;
b = 1;
x[t_] = 
        a Cos[ ((ω  π)/32) t];
y[t_]  =  
        b Sin[((ω  π)/32) t];
ShowAnimation[
        Table[Graphics[{AbsolutePointSize[15], RGBColor[1, 1, 0], 
              Point[{x[m], y[m]}]}, AspectRatio -> Automatic, 
            Background -> RGBColor[0, 0, 0], 
            PlotRange -> {{-1.1 a, 1.1  a}, {-1.1 b, 
                  1.1  b}}], {m, 0, 63}]];

Parametrisation of an elliptic spiral (with variable angular speed)

\(\(\(
\)\(\(ω\  = \ 2;\)

    \(a\  = \ 2;\)

    \(b = 1;\)

    \(x[t_] = a\ Cos[\ \(ω\ \ π\)\/32\ t];\)

    \(y[t_]\  = \ 
        b\ Sin[\(\(ω\ \ π\)\/32\) t];\)

    \(z[t_] = \(t - 32\)\/32;\)

    \(arcs[
          m_] = {{x[t], y[t], z[t]}, {x[t +  .05/ω], 
              y[t +  .05/ω], z[t +  .05/ω]}} /. 
          t -> m;\)

    \(p1 = 
        ShowAnimation[
          Table[Graphics3D[{AbsoluteThickness[5], RGBColor[1, 1, 0], 
                Line[arcs[m]]}, AspectRatio -> Automatic, 
              Background -> RGBColor[0, 0, 0], 
              PlotRange -> {{\(-1.1\) a, 1.1  a}, {\(-1.1\) b, 
                    1.1  b}, {1, \(-1\)}}], {m, 0, 
              63}]];\)


    \)\)\)

ω  =  4;
a  =  1;
b = 1;
x[t_] = 
        a Cos[ ((ω  π)/32) t];
y[t_]  =  
        b Sin[((ω  π)/32) t];
z[t_] = ((t - 32)/32);
arcs = 
        Flatten[Table[{{x[t], y[t], z[t]}, {x[t + 0.05/ω], 
                  y[t + 0.05/ω], z[t +  .05/ω]}} /. 
              t -> m, {m, 0, 63}], 1];
Show[
        Table[Graphics3D[{AbsoluteThickness[3], RGBColor[1, 1, 0], 
              Line[arcs]}, AspectRatio -> Automatic, 
            Background -> RGBColor[0, 0, 0]], {m, 0, 
            63}]];