A problem about the implementation of Bezier curve

Question

I implemented the Bezier curve like built-in BezierCurve as follow:

SplineDegree -> d specifies that the underlying polynomial basis should have maximal degree d.

Method is a option that determining which algorithm was used.

ControlPoints is a option that determining whether show the control points

SplineClosed is an option that specifies whether spline curves or surfaces should be closed.

CAGDBezierCurve by default represents a composite cubic Bézier curve.

With SplineDegree -> d, CAGDBezierCurve with d + 1 control points yields a simple degree - d Bézier curve.With fewer control points, a lower - degree curve is generated. With more control points, a composite Bézier curve is generated.

---

UPDATE for 3D CASE

Options[CAGDBezierCurve] = {SplineClosed -> False, 
 SplineDegree -> Automatic, ControlPoints -> False, Method -> Automatic};

CAGDBezierCurve[pts : {{_, _} ..} | {{_, _, _} ..}, opts : OptionsPattern[]] :=
 Module[{sc, sd, cp, Bezier, ptgroup},
  sc = OptionValue[SplineClosed];
  sd = OptionValue[SplineDegree] /. Automatic -> 3;
  cp = OptionValue[ControlPoints];
  Bezier =
   ToExpression@OptionValue[Method] /. Automatic -> BezierDefinition;
  ptgroup = Partition[pts, sd + 1, sd, 1, {}];
  If[Length@First@pts == 2,
   ParametricPlot[
    Evaluate[Bezier[#, u] & /@ ptgroup], {u, 0, 1},
    Evaluate@
     (Sequence @@
       FilterRules[{opts}, Options[ParametricPlot]]), 
      Axes -> False, PlotRange -> All,
    Epilog -> If[cp, {Green, Line[pts], Red, Point[pts]}, {}]],
   Show[
    {ParametricPlot3D[
      Evaluate[Bezier[#, u] & /@ ptgroup], {u, 0, 1},
      Evaluate@
       (Sequence @@FilterRules[{opts}, Options[ParametricPlot3D]]), 
      Axes -> False, PlotRange -> All],
    Graphics3D[
     If[cp, {Green, Line[pts], Red, Point[pts]}]]}]
  ]
]

---

BezierDefinition[pts_, u0_?NumericQ] :=
 Nest[
  MovingAverage[
   ArrayPad[#, 1], {u0, 1 - u0}] &, {1}, Length[pts] - 1].pts

deCasteljau[pts_, u0_?NumericQ] :=
 Nest[
  MovingAverage[#, {1 - u0, u0}] &, pts, Length@pts - 1]

Owning to the HoldAll attribute of ParametricPlot, so I must used the Evaluate three times in the ParametricPlot.

In addition, I know the ParametricPlot owns the Evaluated-> True option, so I refactor(remove Evaluate) the part of ParametricPlot as below:

ParametricPlot[
 Bezier[#, u] & /@ ptGroup, {u, 0, 1},
 Evaluated -> True,
 Sequence @@
  FilterRules[{opts}, Options[ParametricPlot]],
 Axes -> False, PlotRange -> All, 
 Epilog -> If[cp, {Green, Line[pts], Red, Point[pts]}, {}]
]

However, it failed.

TEST

pts = {{0, 0}, {1, 1}, {2, -1}, {3, 0}, {4, -2}, {5, 1}};

CAGDBezierCurve[pts, SplineDegree -> #1, SplineClosed -> #2, 
 ControlPoints -> True] & @@@ {{2, True}, {3, False}} // Row

Graphics[{Green, Line[pts], Red, Point[pts], Black, 
 BezierCurve[pts, SplineDegree -> #1, 
  SplineClosed -> #2]}] & @@@ {{2, True}, {3, False}} // Row

pts1 = {{0, 0, 0}, {1, 1, 1}, {2, -1, 1}, {3, 0, 2}, {5, 3, 4}};

CAGDBezierCurve[pts1, SplineDegree -> 4, ControlPoints -> True]

---

QUESTION

• Is there a solution to deal with avoiding using Evaluate function many times?

• Why the PlotRange -> All cannot show the entire graph?

Answer 1

I have reworked your code somewhat. I hope what I have done will help you with your problem.

BezierDefinition[pts_, u0_?NumericQ] := 
  Nest[MovingAverage[ArrayPad[#, 1], {u0, 1 - u0}] &, {1}, Length[pts] - 1].pts

ClearAll @ CAGDBezierCurve;
Options[CAGDBezierCurve] = 
  {SplineClosed -> False, SplineDegree -> Automatic, 
   ControlPoints -> False, Method -> Automatic};
CAGDBezierCurve[
     pts : {{_, _} ..}, 
     opts : OptionsPattern[{CAGDBezierCurve, ParametricPlot}]] := 
   Module[{sc, sd, cp, Bezier, ptGroup},
   sc = OptionValue[SplineClosed];
   sd = OptionValue[SplineDegree] /. Automatic -> 3;
   cp = OptionValue[ControlPoints];
   Bezier = 
    ToExpression @ OptionValue[Method] /. Automatic -> BezierDefinition;
   ptGroup = 
     If[sc, 
        Partition[Append[pts, First@pts], sd + 1, sd, 1, {}], 
        Partition[pts, sd + 1, sd, 1,
        {}]];
   ParametricPlot[Bezier[#, u] & /@ ptGroup, {u, 0, 1}, 
     Evaluate @ FilterRules[{opts}, Options[ParametricPlot]],
     Axes -> False, PlotRange -> All, 
     Epilog -> If[cp, {Green, Line[pts], Red, Point[pts]}, {}]]]

CAGDBezierCurve[pts,
  SplineDegree -> #1, SplineClosed -> #2, ControlPoints -> True, 
  ImageSize -> Medium, PlotRange -> {{0., 5.}, {-2., 1.}}, 
  PlotRangePadding -> {.25, .3}] & @@@ {{2, True}, {3, False}} // Column

Discussion

1. I have reduced the number of Evaluates to one; the others were simply not necessary, at least in V10.2. Over the last several releases Mathematica's plotting functions have become much smarter about evaluating held arguments internally.

2. I have corrected mistakes made in handling options. The documentation on option handling is a disgrace, so don't feel bad about not getting it right.

3. I have removed your alternative pattern for 3D points; there is no way your function as currently written can handle such points.

4. I have not done anything about the clipping of Epilog graphics. The example output shown above demonstrates the missing graphics can be revealed by explicitly giving the plot range and increasing the plot range padding. This is a work-around, not a solution. Unfortunately, PlotRange -> All does not look at the epilog graphics when computing the plot range.

I know of no easy fix for the plot range problem. You might try to compute the needed space in your function and set the plot range explicitly.

Update

ShutaoTang has posted an answer with incorporates the points made above and also extends his CAGDBezierCurve function to handle 3D control points. I believe I can improve on his code a little by consolidating the two calls to Show into a single call simply by recognizing that the heads of expressions can be variables that get evaluated in the same way arguments do.

ClearAll@CAGDBezierCurve;
Options[CAGDBezierCurve] = 
  {SplineClosed -> False, SplineDegree -> Automatic, ControlPoints -> False, 
   Method -> Automatic};
CAGDBezierCurve[
    pts : {{_, _} ..} | {{_, _, _} ..}, 
    opts : OptionsPattern[{CAGDBezierCurve, ParametricPlot, ParametricPlot3D}]] :=
  Module[{sc, sd, cp, Bezier, ptgroup},
    sc = OptionValue[SplineClosed];
    sd = OptionValue[SplineDegree] /. Automatic -> 3;
    cp = OptionValue[ControlPoints];
    Bezier = 
      ToExpression @ OptionValue[Method] /. Automatic -> BezierDefinition;
    ptgroup = 
      If[sc, 
        Partition[Append[pts, First@pts], sd + 1, sd, 1, {}], 
        Partition[pts, sd + 1, sd, 1, {}]];
     Block[{u, plotF, graphF},
       With[{curves = Bezier[#, u] & /@ ptgroup},
         {plotF, graphF} =
           If[Length @ First @ pts == 2,
             {ParametricPlot, Graphics}, 
             {ParametricPlot3D, Graphics3D}];
         Show[{
           plotF[curves, {u, 0, 1},
             Evaluate@FilterRules[{opts}, Options[plotF]], Axes -> False, 
             PlotRange -> All], 
           graphF[If[cp, {Green, Line[pts], Red, Point[pts]}]]}]]]]