Off[General::"spell1"];

    
xmin = -2;  xmax = 2;  ymin  = -2;  ymax = 2;

t = 2;

    
Show[
Graphics[{
{PointSize[
                0.02], Blue, 
              Point[{1, 
                  1}]}
}], 
Frame -> 
          True, PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}}, 
AspectRatio -> Automatic,  
        GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10]}
];

t =  .5;

    
Show[
Graphics[{
{PointSize[
                0.02], Blue, 
              Point[{Cos[t], 
                  Sin[t]}]}
}], 
Frame -> True, 
        PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}}, 
AspectRatio -> Automatic,  
        GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10]}
];

Clear[t];  xmin = -2;  xmax = 2;  ymin  = -2;  
    ymax = 2;
Table[
Show[
Graphics[{
{PointSize[0.02], Blue, 
              Point[{Cos[t], Sin[t]}]}
}], 
        PlotLabel -> N[t], 
Frame -> True, 
        PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}}, 
AspectRatio -> Automatic,  
        GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10]}
], {t, 0, 2  Pi, 
        (Pi/32)}]; 
    SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t]; xmin = -2;  xmax = 2;  ymin  = -2;  
    ymax = 2;
Table[
Show[
Graphics[{
{PointSize[0.02], Blue, 
              Table[Point[{Cos[v], Sin[v]}], {v, 0, t, 
                  (Pi/64) }]}
}], 
        PlotLabel -> N[t], 
Frame -> True, 
        PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}}, 
AspectRatio -> Automatic,  
        GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10]}
], {t, 0, 2  Pi, 
        (Pi/64)}]; 
    SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t];  xmin = -2;  xmax = 2;  ymin  = -2;  
    ymax = 2;
Table[
Show[
Graphics[{
{PointSize[0.02], Blue, 
              Table[Point[(1 + Cos[3  v]) {Cos[v], Sin[v]}], {v, 0, t, 
                  (Pi/64) }]}
}], 
Frame -> True, 
        PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}}, 
AspectRatio -> Automatic,  
        GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10]}
], {t, 0, 2  Pi, 
        (Pi/64)}]; 
    SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t]; xmin = -2;  xmax = 2;  ymin  = -2;  
    ymax = 2;
Table[
Show[
Graphics[{
{PointSize[0.02], Blue, 
                Table[Point[(1 + Cos[v]) {Cos[v], Sin[v]}], {v, 0, t, 
                    (Pi/64) }]}
}], 
Frame -> True, 
          PlotRange -> {{xmin, xmax}, {ymin, 
                ymax}}, 
AspectRatio -> Automatic,  
          GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
                Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
                Range[-10, 10]}
], {t, 0, 2  Pi, 
          (Pi/64)}];
SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t]; xmin = -2;  xmax = 2;  ymin  = -2;  
    ymax = 2;
Table[
Show[
Graphics[{
{PointSize[0.02], Blue, 
              Table[Point[(1 + Cos[2  v]^2) {Cos[v], Sin[v]}], {v, 0, t, 
                  (Pi/64) }]}
}], 
Frame -> True, 
        PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}}, 
AspectRatio -> Automatic,  
        GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10]}
], {t, 0, 2  Pi, 
        (Pi/64)}]; 
    SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t, a]; a[t_] := Cos[t]; b[t_] := Sin[t]; 
    t0 = 1;
pP = {a[t0], b[t0]};

 xmin = -2;  xmax = 2;  ymin  = -2;  
    ymax = 2;

Table[
Show[
Graphics[{
{PointSize[0.02], Blue, 
              Table[Point[{a[v], b[v]}], {v, 0, t, (Pi/64) }]}, {PointSize[
                0.02], Red, Point[pP]}, 
{Thickness[0.005], 
              Red, Line[{pP - 10 {a'[t0], b'[t0]}, 
                  pP + 10 {a'[t0], b'[
                          t0]}}]}
}], 
Frame -> True, 
        PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}}, 
AspectRatio -> Automatic,  
        GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10]}
], {t, 0, 2  Pi, 
        (Pi/64)}]; 
    SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t, a]; a[t_] :=  Cos[t]; b[t_] :=  Sin[t]; 
    t0 = 1;
pP = {a[t0], b[t0]};  xmin = -3;  xmax = 3;  ymin  = -3;  
    ymax = 3;  st  = (Pi/64);

Table[
Show[
Graphics[{
{PointSize[0.02], Blue, 
              Table[Point[{a[v], b[v]}], {v, 0, 2 Pi, st}]}, {PointSize[
                0.01], Red, 
              Table[Point[{a[v], b[v]}], {v, 0, t, 
                  st }]}, 
{Thickness[0.004], Red, 
              Table[Line[{ {a[v], b[v]} - 
                      10 {a'[v], b'[v]}, {a[v], b[v]} + 
                      10 {a'[v], b'[v]}}], {v, 0, t, 
                  st }]}
}], 
Frame -> True, 
        PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}}, 
AspectRatio -> Automatic,  
        GridLines -> {{#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10], {#, {GrayLevel[0.8]}} & /@ 
              Range[-10, 10]}
], {t, 0, 2  Pi, 
        (Pi/64)}]; 
    SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t];  xmin = -2;  xmax = 2;  ymin  = -2;  ymax = 2; 
    zmin = 0; zmax = Pi;
Table[
Show[
Graphics3D[{
{PointSize[0.02], Blue, 
              Table[Point[{Cos[v], Sin[v], (v/6)}], {v, 0, t, 
                  (Pi/32)}]}
}], 
        PlotLabel -> 
          N[t], 
PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}, {zmin, zmax}}, 
        Axes ->  True
], {t, 0, 6  Pi, (Pi/32)}]; 
    SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t, a, b, c];  xmin = -2;  xmax = 2;  ymin  = -2;  
    ymax = 2; zmin = 0; zmax = 4;  st = (Pi/32);
a[t_] := Cos[t]; b[t_] := Sin[t]; 
    c[t_] := (t/Pi);

Table[
Show[
Graphics3D[{
{PointSize[0.02], Blue, 
              Table[Point[{a[v], b[v], c[v]}], {v, 0, 4 Pi, 
                  st}]}, 
{Thickness[0.004], Red, 
              Table[Line[{ {a[v], b[v], 
                        c[v]} - (2/Norm)[{a'[v], b'[v], c'[
                                  v]}] {a'[v], b'[v], c'[
                            v]}, {a[v], b[v], 
                        c[v]} + (2/Norm)[{a'[v], b'[v], c'[
                                  v]}] {a'[v], b'[v], c'[
                            v]}}], {v, 0, t, st }]}
}], 
        PlotLabel -> 
          N[t], 
PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}, {zmin, zmax}}, 
        Axes ->  True
], {t, 0, 4  Pi, st}]; 
    SelectionMove[EvaluationNotebook[], All, GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]

Clear[t, a, b, c];  xmin = -2;  xmax = 2;  ymin  = -2;  
    ymax = 2; zmin = 0; zmax = 4;  st = (Pi/64);
a[t_] := (Sqrt[ .1 + (((2 - (t/Pi)))^2)]) Cos[2  t]; 
    b[t_] := (Sqrt[ .1 + (((2 - (t/Pi)))^2)]) Sin[2  t]; 
    c[t_] := (t/Pi);

Table[
Show[
Graphics3D[{
{PointSize[0.02], Blue, 
              Table[Point[{a[v], b[v], c[v]}], {v, 0, 4 Pi, 
                  st}]}, 
{Thickness[0.004], Red, 
              Table[Line[{ {a[v], b[v], 
                        c[v]} - (2/Norm)[{a'[v], b'[v], c'[
                                  v]}] {a'[v], b'[v], c'[
                            v]}, {a[v], b[v], 
                        c[v]} + (2/Norm)[{a'[v], b'[v], c'[
                                  v]}] {a'[v], b'[v], c'[
                            v]}}], {v, 0, t, st }]}
}], 
        PlotLabel -> 
          N[t], 
PlotRange -> {{xmin, xmax}, {ymin, 
              ymax}, {zmin, zmax}}, Axes ->  True, 
        BoxRatios -> {1, 1, 1.5}
], {t, 0, 4  Pi, 
        st}]; SelectionMove[EvaluationNotebook[], All, 
      GeneratedCell];
FrontEndTokenExecute["OpenCloseGroup"]