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Re Bezier curves mapped from 2D to 3D surface

• To: mathgroup at smc.vnet.net
• Subject: [mg124650] Re Bezier curves mapped from 2D to 3D surface
• From: Chris Young <cy56 at comcast.net>
• Date: Sat, 28 Jan 2012 06:36:52 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

I fixed it so the curves are mapped correctly to the saddle surface. I 
forgot that my butterfly function was a set of 2D vectors. Changing the 
"@" to a "/@" fixed everything. However, now it's impossibly slow.

http://home.comcast.net/~cy56/Mma/BezCurveToSurface2.nb

http://home.comcast.net/~cy56/Mma/BezCurveToSurface2Pic.png

DynamicModule[
 {
  \[ScriptCapitalP],            (*
  the points for each Bézier curve *)
  \[ScriptCapitalC],            \
 (* the parametric functions for the Bézier curves *)

  PM,           (* the points for the morphing Bézier curves *)

  CM,           (*
  the parametric functions for the morphing Bézier curves *)

  curveStyle,
  wing, butterfly, butterflyPlot
  },

   (* s is the number of curves *)

 curveStyle[a_, b_, c_, s_, th_] :=  Table[
   {Directive[AbsoluteThickness[th], Hue[vFnc[a, b, c, k]]]},
   {k, 0, 1, 1/s}];
 wing[n_, t_, s_] := Table[CM[n][k, t], {k, 0, 1, 1/s}];
 butterfly[t_, s_] := Join[wing[1, t, s], wing[2, t, s]];

 butterflyPlot[a_, b_, c_, s_, th_] :=
  ParametricPlot[
   Evaluate @ butterfly[t, s - 1],     {t, 0, 1},
   PlotStyle -> curveStyle[a, b, c, 2 s - 1, th]
   ];

 Dynamic @ Manipulate[
   \[ScriptCapitalP][1] = Take[P, {1, 5}];
   \[ScriptCapitalP][2] = Take[P, {6, 10}];
   \[ScriptCapitalP][3] = Take[P, {11, 15}];

   \[ScriptCapitalC][1][
     t_] := \[HorizontalLine]Bez[\[ScriptCapitalP][1],
     t]; \[ScriptCapitalC][2][
     t_] := \[HorizontalLine]Bez[\[ScriptCapitalP][2],
     t]; \[ScriptCapitalC][3][
     t_] := \[HorizontalLine]Bez[\[ScriptCapitalP][3], t];

   PM[1][n_] := (1 - n) \[ScriptCapitalP][1] +
     n \[ScriptCapitalP][2];
   PM[2][n_] := (1 - n) \[ScriptCapitalP][2] +
     n \[ScriptCapitalP][3];
   CM[1][n_, t_] := \[HorizontalLine]Bez[PM[1][n], t];
   CM[2][n_, t_] := \[HorizontalLine]Bez[PM[2][n], t];

   Grid[
    {
     {
      LocatorPane[
       Dynamic @ P,

       Dynamic @ Show[
         butterflyPlot[a, b, c, nCurves, thick],
         Graphics @ {Gray, Dashed, Line[\[ScriptCapitalP][1]],
           Line[\[ScriptCapitalP][2]], Line[\[ScriptCapitalP][3]]},
         PlotRange -> 2,
         Axes -> True,
         ImageSize -> 4* 72
         ],
       {{-2, -2}, {2, 2}, {.25, .25}}
       ],

      Dynamic @
       Show[
        ParametricPlot3D[
         \[HorizontalLine]Saddle @ {u, v}, {u, -2, 2}, {v, -2, 2},
         PlotStyle -> Opacity[opac],
         Mesh -> False
         ],
        ParametricPlot3D[
          \[HorizontalLine]Saddle /@  butterfly[t, nCurves], {t, 0,
           1}
          ] /.
         Line[pts_, rest___] :> Tube[pts, 0.05, rest],
        PlotRange -> 2,
        ImageSize -> 4* 72
        ]
      }
     }
    ],
   {{P,
     {
      {-1, -2}, {-1, -1}, {-1, 0}, {-1, 1}, {-1, 2},
      {  0, -2}, {   0, -1}, {   0, 0}, {  0, 1}, {   0, 2},
      {  1, -2}, {   1, -1}, {  1, 0}, {   1, 1}, {  1, 2}
      }
     }, None},

   {{nCurves, 5}, 1, 20, 1},        (*
   one less than the number of "morphing" curves *)
   {{thick, 12},
    1, 30, 1},
   (* start, end, and middle hues *)
   {{a, 0.1},  0, 1},
   {{b, 0.5}, 0, 1},
   {{c, 0.1}, 0, 1},

   {{opac, 0.75}, 0, 1}
   ]
 ]