Examples of curves

A line segment in R^2

gamma[t_] = t{1,2}+(1-t){-1,0}
ParametricPlot[gamma[t],{t,0,1}]

A semicircle (of radius 2, centered at (1,1))

gamma[t_]={2Cos[t],2Sin[t]}+{1,1}
ParametricPlot[gamma[t],{t,-Pi/2,Pi/2}]

A cusp curve

ParametricPlot[{t^2,t^3},{t,-1/2,1/2}]

A Helix

ParametricPlot3D[{Sin[t],Cos[t],t/10},{t,-10,10}]

3 D Spirograph

gamma[t_]=3*Sin[13t]{1,0,0}+5*Cos[11t]{0,1,0}+4*Sin[9t]{0,0,1}
ParametricPlot3D[gamma[t],{t,-100,100},PlotPoints->10000]