\(as = u; 
bs = \(-2\)\ u\ v\  - \ 4\ v^3; 
cs = \ 3\ v^4\  + \ u\ v^2; 
    \)

swallowtail = 
      ParametricPlot3D[{as, bs, cs -  .01}, {v, -1, 1}, {u, -3, 3}]

curve = ParametricPlot3D[{-2 v^2, 0, v^4}, {v, 0, 1}]

Show[swallowtail, curve]

\(For[r = 0; n = 0, r <= \ 1.0, r +=  .05; n += 1, 
	b = 0; 
      a = \(-2\)\ r^2; c = r^4; 
	
      p = Graphics3D[{PointSize[ .04], RGBColor[1, 0, 0], Point[{a, b, c}]}]; 
      
	pic[n] = 
        GraphicsArray[{
            Show[{p, curve, swallowtail2}, AspectRatio -> Automatic, 
              Boxed -> False, ViewPoint -> {\(-1\), \(-3\), 4}], 
            Plot[x^4\  + \ a\ x^2\  + \ b\ x\  + c, {x, \(-2\), 2}, 
              PlotRange -> {{\(-3\), 3}, {\(-3\), 3}}, Ticks -> None, 
              AxesLabel -> {"x", "p(x)"}]}]]\)

\(For[r = 0; n = 0, r <= \ 1.0, r +=  .05; n += 1, 
	b = 0; 
      a = \(-2\)\ r^2; c = r^4; 
	
      p = Graphics3D[{PointSize[ .04], RGBColor[1, 0, 0], Point[{a, b, c}]}]; 
      
	pic[n] = 
        GraphicsArray[{
            Show[{p, curve, swallowtail2}, AspectRatio -> Automatic, 
              Boxed -> False, ViewPoint -> {\(-1\), \(-3\), 4}], 
            Plot[x^4\  + \ a\ x^2\  + \ b\ x\  + c, {x, \(-2\), 2}, 
              PlotRange -> {{\(-3\), 3}, {\(-3\), 3}}, Ticks -> None, 
              AxesLabel -> {"x", "p(x)"}]}]]\)

\(\(	\(For[r = 1; n = 21, r >= \ 0, r += \(- .05\); n += 1, 
	b = 0; 
      a = \(-2\)\ r^2; 
      c = r	For[r = 1; n = 21, r >= \ 0, r += \(- .05\); n += 1, 
	b = 0; 
            a = \(-2\)\ r^2; c = r^4; 
	
            p = Graphics3D[{PointSize[ .04], RGBColor[1, 0, 0], 
                  Point[{a, b, c}]}]; 
	
            pic[n] = 
              GraphicsArray[{
                  Show[{p, curve, swallowtail2}, AspectRatio -> Automatic, 
                    Boxed -> False, ViewPoint -> {\(-1\), \(-3\), 4}], 
                  Plot[x^4\  + \ a\ x^2\  + \ b\ x\  + c, {x, \(-2\), 2}, 
                    PlotRange -> {{\(-3\), 3}, {\(-3\), 3}}, Ticks -> None, 
                    AxesLabel -> {"x", "p(x)"}]}]]
\)\)\)

n = 1; 
    Do[Display[StringJoin["picket.", ToString[10 + n]], pic[n], 
        "GIF", ImageSize -> {600, 371}], {n, 1, 40}];