(* Below are plots of the paraboloid, cone, and saddle, done both with Cartesian grids
and with a polar coordinate grid--so you can see radial symmetry more clearly. Evaluate the notebook. You can then grab and rotate figures in real time with your mouse.

At the bottom is a "movie" of the cycloid. *)

Plot3D[x^2+y^2,{x,-2,2},{y,-2,2}, Boxed->False,AxesOrigin->{0,0,0}]
ParametricPlot3D[{r Cos[t], r Sin[t], r^2},{r,0,2},{t,0,2Pi}, Boxed->False,AxesOrigin->{0,0,0}]

Plot3D[Sqrt[x^2+y^2],{x,-2,2},{y,-2,2}, Boxed->False,AxesOrigin->{0,0,0}]
ParametricPlot3D[{r Cos[t], r Sin[t], r},{r,0,2},{t,0,2Pi}, Boxed->False,AxesOrigin->{0,0,0}]

Plot3D[-x^2+y^2,{x,-2,2},{y,-2,2}, Boxed->False,AxesOrigin->{0,0,0}]
ParametricPlot3D[{r Cos[t], r Sin[t], r^2 Cos[2t]},{r,0,2},{t,0,2Pi}, Boxed->False,AxesOrigin->{0,0,0}]

cycloid=ParametricPlot[{t- Sin[t],1- Cos[t]},{t,0,4 Pi}];
pt[s_]:=Graphics[{RGBColor[1,0,0],PointSize[0.02],Point[{s-Sin[s],1- Cos[s]}//Evaluate]}];
circ[s_]:=ParametricPlot[{s+Cos[t],1+Sin[t]}//Evaluate,{t,0,2Pi}];
seg[s_]:=Graphics[Line[{{s,1},{s-Sin[s],1- Cos[s]}}//Evaluate]];
Animate[
Show[cycloid,pt[s],circ[s],seg[s],AspectRatio->Automatic,Ticks->None,AxesOrigin->{0,0}],{s,0,4 Pi}]