Parametric Plots

Ex 1

Circ1[a_,b_]:=Manipulate[
ParametricPlot[{Cos[t],Sin[t]}, {t,a,k}, 
ImageSize -> 275, PlotRange -> {{-1.1, 1.1}, {-1.1, 1.1}}, AspectRatio -> 1, Ticks -> Automatic], {k, a+.01,b}];Circ1[0,2*Pi]

Ex 2

Circ2[a_,b_]:=Manipulate[
ParametricPlot[{Sin[2t],Cos[2t]}, {t,a,k}, 
ImageSize -> 275, PlotRange -> {{-1.1, 1.1}, {-1.1, 1.1}}, AspectRatio -> 1, Ticks -> Automatic], {k, a+.01,b}];Circ2[0,2*Pi]

Ex 3

Curv3[a_,b_]:=Manipulate[
ParametricPlot[{t^3-t,t^2}, {t,a,k}, 
ImageSize -> 275, PlotRange -> {{-1, 6}, {-0.1,4 }}, AspectRatio -> 1, Ticks -> Automatic], {k, a+.01,b}];Curv3[0,2]

Ex 4 (Line)

Curv4[a_,b_]:=Manipulate[
ParametricPlot[{2*t + 1,t- 2}, {t,a,k}, 
ImageSize -> 275, PlotRange -> {{-30, 30}, {-30,30 }}, AspectRatio -> 1, Ticks -> Automatic], {k, a+.01,b}];Curv4[-10,10]

Ex 5

Circ1[a_,b_]:=Manipulate[
ParametricPlot[{2Cos[t],3Sin[t]}, {t,a,k}, 
ImageSize -> 275, PlotRange -> {{-3.1, 3.1}, {-3.1, 3.1}}, AspectRatio -> 1, Ticks -> Automatic], {k, a+.01,b}];Circ1[0,4*Pi]

Extra Ex

extra[a_,b_]:=Manipulate[
ParametricPlot[{Sin[t],(Cos[t])^2}, {t,a,k}, 
ImageSize -> 275, PlotRange -> {{-1.1, 1.1}, {-1.1, 1.1}}, AspectRatio -> 1, Ticks -> Automatic], {k, a+.01,b}];extra[Pi/2,2*Pi]

Cycloid

cycloid[opts_Rule]:=Manipulate[Show[ParametricPlot[{t-Sin[t], 1-Cos[t]},{t, -3, k}, PlotRange ->{{-1, 30.5},{0, 2.1}},opts,  PlotStyle -> Thick, PlotRangePadding -> .1],
Graphics[{Table[Line[{{k, 1}, {k + Cos[θ-k], 1+Sin[θ-k]}}], {θ, -3Pi/8, 11Pi/8, Pi/8}], Thickness[Medium], 
Circle[{k, 1},1], PointSize[Medium], Red,
Line[{{k, 1},  {k + Cos[Pi/2+k], 1-Sin[Pi/2+k]}}],
Point[{k - Sin[k], 1-Cos[k]}] }], Ticks -> None], {k, -2, 10Pi+2}]

cycloid[ImageSize->670]

Lissajous

Liss[a_,n_,b_,m_,t1_,t2_]:=
Manipulate[
ParametricPlot[{a*Sin[n*t],b*Cos[m*t]}, {t,t1,k}, 
ImageSize -> 275, PlotRange -> {{-3, 3}, {-10, 10}}, AspectRatio -> 1, Ticks -> Automatic], {k, t1+.01, t2}];Liss[2, 3, 5, 7, 0,2Pi]

Butterfly

Bu[a_,b_]:=
Manipulate[
ParametricPlot[{Sin[t](Exp[Cos[t]] - 2Cos[4t] - (Sin[t/12])^5),Cos[t](Exp[Cos[t]] - 2Cos[4t] - (Sin[t/12])^5)},{t,a,k},  ImageSize -> 275, PlotRange -> {{-5, 5}, {-5, 5}}, AspectRatio -> 1, Ticks -> Automatic], {k, a+.01, b}];Bu[0, 10*Pi]

Spiral

Manipulate[
ParametricPlot[{t*Sin[t],t*Cos[t]}, {t,0,k}, 
ImageSize -> 275, PlotRange -> {{-10, 10}, {-10, 10}}, AspectRatio -> 1, Ticks -> Automatic], {k,.01,3Pi}]

Manipulate[
ParametricPlot[{t*Cos[t],t*Sin[t]}, {t,0,k}, 
ImageSize -> 275, PlotRange -> {{-10, 10}, {-10, 10}}, AspectRatio -> 1, Ticks -> Automatic], {k,.01,3Pi}]

Hypercycloid

Hyp[a_,b_,t1_,t2_]:=
Manipulate[
ParametricPlot[{(a-b)*Cos[t]+b*Cos[((a-b)/b)*t],(a-b)*Sin[t]-b*Sin[((a-b)/b)*t]}, {t,t1,k}, 
ImageSize -> 275, PlotRange -> {{-8, 8}, {-8, 8}}, AspectRatio -> 1, Ticks -> Automatic], {k, t1+.01, t2}];Hyp[4, 1,  0,100*Pi]

Spirographs

Spi[a_,b_,p_,t1_,t2_]:=
Manipulate[
ParametricPlot[{(a-b)*Cos[t]+p*b*Cos[((a-b)/b)*t],(a-b)*Sin[t]-p*b*Sin[((a-b)/b)*t]}, {t,t1,k}, 
ImageSize -> 275, PlotRange -> {{-2, 2}, {-2, 2}}, AspectRatio -> 1, Ticks -> Automatic], {k, t1+.01, t2}];Spi[11/7, 1,.5,  0,20*Pi]