Toroidal Spiral

Equation

Edit the equation below to change the animation below.
(You will have to re-execute the Animate command to see the changes.)

r[t_] := {(2+Sin[3 t]) Cos[t], (2+Sin[3 t]) Sin[t], Cos[3 t]};

Animation Code

frames[time_,showVector_]:=(v=r[time];point={Blue,PointSize[0.02],Point[v]};vector=If[showVector,{Red,Thickness[0.008],Arrow[{{0,0,0},v}]},{}];Show[Graphics3D[{point,vector,Text[x,{3.2,0,0}],Text[y,{0,3.2,0}],Text[z,{0,0,3.2}]}],ParametricPlot3D[r[s],{s,0,time},PlotStyle->{Orange,Thickness[0.008]}],Axes->True,AxesOrigin->{0,0,0},PlotRange->{{-3,3},{-3,3},{-3,3}},Boxed->False,PlotLabel->Style[Row[{"The toroidal spiral\n",x[t]==r[t][[1]],"\n",y[t]==r[t][[2]],"\n",z[t]==r[t][[3]]}],FontSize->16],ViewPoint->{2000,400,800}])

Animation

Change showVector to True or False to toggle displaying the vector as the animation traces the curve.
(You will have to re-execute the Animate command to see the changes.)

showVector=True;
Animate[frames[t,showVector],{t,0,2 Pi}]

Twisted Cubic

Equation

Edit the equation below to change the animation below.
(You will have to re-execute the Animate command to see the changes.)

r[t_]:={4 t,2 t^2, t^3};

Animation Code

frames[time_,showVector_]:=(v=r[time];point={Blue,PointSize[0.02],Point[v]};vector=If[showVector,{Red,Thickness[0.008],Arrow[{{0,0,0},v}]},{}];Show[Graphics3D[{point,vector,Text[x,{8.2,0,0}],Text[y,{0,8.2,0}],Text[z,{0,0,8.2}]}],ParametricPlot3D[r[s],{s,-2,time},PlotStyle->{Orange,Thickness[0.008]}],Axes->True,AxesOrigin->{0,0,0},PlotRange->{{-8,8},{-8,8},{-8,8}},Boxed->False,PlotLabel->Style[Row[{"The twisted cubic\n",x[t]==r[t][[1]],"\n",y[t]==r[t][[2]],"\n",z[t]==r[t][[3]]}],FontSize->16],ViewPoint->{2000,400,800}])

Animation

Change showVector to True or False to toggle displaying the vector as the animation traces the curve.
(You will have to re-execute the Animate command to see the changes.)

showVector=True;
Animate[frames[t,showVector],{t,-2,2}]

Nonsmooth Curve

Equation

Edit the equation below to change the animation below.
(You will have to re-execute the Animate command to see the changes.)

r[t_]:={t^4, 2t^3, 4t^2};

Animation Code

frames[time_,showVector_]:=(v=r[time];point={Blue,PointSize[0.02],Point[v]};vector=If[showVector,{Red,Thickness[0.008],Arrow[{{0,0,0},v}]},{}];Show[Graphics3D[{point,vector,Text[x,{17,0,0}],Text[y,{0,17,0}],Text[z,{0,0,17}]}],ParametricPlot3D[r[s],{s,-2,time},PlotStyle->{Orange,Thickness[0.008]}],Axes->True,AxesOrigin->{0,0,0},PlotRange->{{-16,16},{-16,16},{-16,16}},Boxed->False,PlotLabel->Style[Row[{"The nonsmooth curve\n",x[t]==r[t][[1]],"\n",y[t]==r[t][[2]],"\n",z[t]==r[t][[3]]}],FontSize->16],ViewPoint->{2000,400,800}])

Animation

Change showVector to True or False to toggle displaying the vector as the animation traces the curve.
(You will have to re-execute the Animate command to see the changes.)

showVector=True;
Animate[frames[t,showVector],{t,-2,2}]

Different Parameterizations

Equation

Edit the equations below to change the animation below.
(You will have to re-execute the Animate command to see the changes.)

r1[t_] := {(2+Sin[3 t]) Cos[t], (2+Sin[3 t]) Sin[t], Cos[3 t]};
r2[t_] := r1[2 t];

Animation Code

frames[time_]:=(v1=r1[time];
v2 = r2[time];
point1={PointSize[0.02],Point[v1]};
point2 ={PointSize[0.02],Point[v2]};
vector1 = {Red,Thickness[0.008],Arrow[{{0,0,0},v1}]};
vector2 = {Blue,Thickness[0.008],Arrow[{{0,0,0},v2}]};
Show[Graphics3D[{point1,point2,vector1,vector2,Text[x,{3.2,0,0}],Text[y,{0,3.2,0}],Text[z,{0,0,3.2}]}],ParametricPlot3D[r1[s],{s,0,2 Pi},PlotStyle->{Orange,Thickness[0.008]}],Axes->True,AxesOrigin->{0,0,0},PlotRange->{{-3,3},{-3,3},{-3,3}},Boxed->False,PlotLabel->Style[Row[{"Two parameterizations of the same curve"}],FontSize->16],ViewPoint->{2000,400,800}])

Animation

Animate[frames[t],{t,0,2 Pi}]