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Re: Color Fill areas in 2D graphic
*To*: mathgroup at smc.vnet.net
*Subject*: [mg23403] Re: [mg23354] Color Fill areas in 2D graphic
*From*: Murray Eisenberg <murray at math.umass.edu>
*Date*: Sat, 6 May 2000 02:26:45 -0400 (EDT)
*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst
*References*: <8etpii$n24@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
There's something missing in the first section of the notebook: The
cell
Show[Graphics[
{LightBlue, (curve ParametricDraw[
Evaluate[ellipse[\[Pi]/4, t]], {t, 0, 2\[Pi]}]) /.
Line -> Polygon, Black, curve}],
AspectRatio -> Automatic, PlotRange -> {{-10, 10}, {-10, 10}},
Background -> Linen,
{}];
refers to a symbol "curve" that has not been defined.
David Park wrote:
>
> >
> > Hi,
> > I want combine several ParametricPlot Images. After this is done, how
> > can I fill areas within the image with a specific color ?
> > thanks
> > Roland Pabel
> >
>
> Hi Roland,
>
> If you wish to try out my DrawingPaper routines, which are available at my
> web site below, then the attached Mathematica notebook, ColoredArea.nb,
> shows how to fill in areas with color. (MathGroup readers may contact me for
> a copy of the notebook.)
>
> If you have a closed curve defined by a parametric expression, then all you
> have to do is convert the Line primitive to a Polygon primitive and supply a
> color.
>
> If the area is defined by the combination of several parametric arcs, then
> DrawingPaper has a routine, StitchLineSegments which will stitch the arcs
> together into one line which can then be converted to a Polygon.
>
> Of course, the standard way to color the area between two curves is to use
> FilledPlot, but this requires the first curve to always be above the second
> curve. Sometimes this does not happen in the x-y plane but does happen in a
> different coordinate system. In those cases the DrawingPaper routine
> DrawingTransform can convert a FilledPlot back to x-y coordinates. For
> example, the area outside a unit circle but inside a cardioid can be filled
> with FilledPlot in a theta-r representation and then transformed to a x-y
> representation.
>
> The notebook has examples of coloring in a tilted ellipse, coloring in the
> area bounded by two intersecting parabolas with their axes in the y and -y
> direction, and the circle-cardioid example mentioned above.
>
> The packages you will need for the notebook from my web site are
> DrawingPaper.m and FilledDrawing.m. The packages should be put in the
> AddOns/ExtraPackages/Graphics folder. There are also other Drawing packages
> and a tutorial.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
>
> (***********************************************************************
>
> Mathematica-Compatible Notebook
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> (*CacheID: 232*)
>
> (*NotebookFileLineBreakTest
> NotebookFileLineBreakTest*)
> (*NotebookOptionsPosition[ 8299, 242]*)
> (*NotebookOutlinePosition[ 8952, 265]*)
> (* CellTagsIndexPosition[ 8908, 261]*)
> (*WindowFrame->Normal*)
>
> Notebook[{
>
> Cell[CellGroupData[{
> Cell["Filling Areas with Color", "Title"],
>
> Cell["\<\
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/\
> \>", "Subtitle"],
>
> Cell[TextData[{
> "The Graphics`DrawingPaper` packages can be downloaded from my web
> site. It \
> should be placed in the AddOns/ExtraPackages/Graphics directory. It can
> then \
> be used just like the standard packages. Here is a click-on link to the
> web \
> site:\n",
> ButtonBox["http://home.earthlink.net/~djmp/ ",
> ButtonData:>{
> URL[ "http://home.earthlink.net/~djmp/ "], None},
> ButtonStyle->"Hyperlink"],
> " "
> }], "Text"],
>
> Cell[BoxData[{
> \(Needs["\<Graphics`DrawingPaper`\>"]\), "\[IndentingNewLine]",
> \(Needs["\<Graphics`FilledDrawing`\>"]\), "\[IndentingNewLine]",
> \(Needs["\<Geometry`Rotations`\>"]\)}], "Input"],
>
> Cell[CellGroupData[{
>
> Cell["For a closed curve", "Section"],
>
> Cell["\<\
> This is the parametric representation of an ellipse with 7:3 axes ratio
> and \
> rotated an an angle \[Theta].\
> \>", "Text"],
>
> Cell[BoxData[
> \(ellipse[\[Theta]_, t_] :=
> RotationMatrix2D[\(-\[Theta]\)] . {7 Cos[t], 3 Sin[t]}\)],
> "Input"],
>
> Cell["\<\
> We set the color to be LightBlue (DrawingPaper automatically loads in \
> Graphics`Colors`), parametrically draw the curve, save it, convert the
> Line \
> to a Polygon, change the color to Black and display the curve.\
> \>", "Text"],
>
> Cell[BoxData[
> \(\(Show[
> Graphics[\n\t\t{LightBlue, \((curve
> ParametricDraw[
> Evaluate[ellipse[\[Pi]/4, t]], {t, 0, 2 \[Pi]}])\)
> /.
> Line \[Rule] Polygon, Black, curve}], \n\n
> AspectRatio \[Rule] Automatic,
> PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}},
> Background \[Rule] Linen, \n{}];\)\)], "Input",
> GeneratedCell->False]
> }, Closed]],
>
> Cell[CellGroupData[{
>
> Cell["For a region defined by several arcs", "Section"],
>
> Cell["\<\
> Here are two parabolas with their axes in the y and -y directions. (If
> they \
> were in the x direction we could use FilledPlot or FilledDraw.)\
> \>", "Text"],
>
> Cell[BoxData[{
> \(\(par1[t_] := {\(-3\) + t\^2, t};\)\), "\[IndentingNewLine]",
> \(par2[t_] := {3 - t\^2, t}\)}], "Input"],
>
> Cell["This finds their points of intersection.", "Text"],
>
> Cell[CellGroupData[{
>
> Cell[BoxData[
> \(Solve[par1[t] \[Equal] par2[t]]\)], "Input"],
>
> Cell[BoxData[
> \({{t \[Rule] \(-\ at 3\)}, {t \[Rule] \ at 3}}\)], "Output"]
> }, Open ]],
>
> Cell["This plots the two curves to show the area we wish to color.",
> "Text"],
>
> Cell[BoxData[
> \(\(Show[
> Graphics[\n\t\t{ParametricDraw[
> Evaluate[{par1[t], par2[t]}], {t, \(-\ at 3\), \ at 3}]}], \n\n
> AspectRatio \[Rule] Automatic,
> PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}/2,
> Background \[Rule] Linen, \n{}];\)\)], "Input",
> GeneratedCell->False],
>
> Cell["\<\
> We want to define a curve by stitching the two arc segments together.
> Then \
> all we have to do is convert the Line to a Polygon. DrawingPaper
> contains a \
> routine for stitching together arcs. The only thing we have to be
> careful \
> about is stitching them end to end in the correct order. Since the
> plotting \
> iterators always run from the minimum value to the maximum value (no
> matter \
> what order we put them), we will often have to reverse the order of the
> \
> points in some of the line segments.\
> \>", "Text"],
>
> Cell[CellGroupData[{
>
> Cell[BoxData[
> \(\(?StitchLineSegments\)\)], "Input"],
>
> Cell[BoxData[
> \("StitchLineSegments[segs_List] will stitch together a list of line
> \
> segments into one line segment. One must be careful with the order of
> the \
> segments and the order of the points within each segment. The segs list
> is \
> Flattened as the first step."\)], "Print"]
> }, Open ]],
>
> Cell["\<\
> So here we assemble our curve by drawing the two line segments
> separately and \
> stitching them together.\
> \>", "Text"],
>
> Cell[BoxData[
> \(\(curve
> StitchLineSegments[{ParametricDraw[
> par1[t] // Evaluate, {t, \(-\ at 3\), \ at 3}],
> ParametricDraw[par2[t] // Evaluate, {t, \(-\ at 3\), \ at 3}] /.
> Line[a_] \[RuleDelayed] Line[Reverse[a]]}];\)\)],
> "Input"],
>
> Cell["\<\
> Now it is a simple matter to draw the colored area and outline it in
> Black.\
> \>", "Text"],
>
> Cell[BoxData[
> \(\(Show[
> Graphics[\n\t\t{LightBlue, curve /. Line \[Rule] Polygon, Black,
>
> curve}], \n\nAspectRatio \[Rule] Automatic,
> PlotRange \[Rule] {{\(-10\), 10}, {\(-10\), 10}}/2,
> Background \[Rule] Linen, \n{}];\)\)], "Input",
> GeneratedCell->False]
> }, Closed]],
>
> Cell[CellGroupData[{
>
> Cell["\<\
> For a region where FilledPlot can be used in a different coordinate
> system\
> \>", "Section"],
>
> Cell["\<\
> Sometimes we can use a different coordinate system where FilledPlot will
> work \
> for the filling of the areas between two curve. For example, to color
> the \
> area inside of a cardioid, but outside a unit circle, we can do a
> FilledDraw \
> in the \[Theta]-r plane. Then we can use the DrawingTransform routine
> from \
> DrawingPaper to transform to an x-y representation.\
> \>", "Text"],
>
> Cell[CellGroupData[{
>
> Cell[BoxData[
> \(\(?DrawingTransform\)\)], "Input"],
>
> Cell[BoxData[
> \("DrawingTransform[f1, f2], where f1 and f2 are function names, is
> a \
> rule which applies the following transformation to all points in the
> graphics \
> object: {x_?NumberQ, y_?NumberQ} \[Rule] {f1[x, y], f2[x, y]}."\)],
> "Print"]
> }, Open ]],
>
> Cell[TextData[{
> "This shows how ",
> Cell[BoxData[
> \(DrawingTransform\)]],
> " can often be used to simplify the filling of areas. We make an
> \[Theta]-r \
> plot and then transform to rectangular coordinates. Don't mix up the
> order of \
> \[Theta] and r! DrawPolarR is the DrawingPaper version of PolarPlot and
> is \
> used to outline the regions."
> }], "Text"],
>
> Cell[BoxData[
> \(\(Show[
> Graphics[\n\t\t{FilledDraw[{1,
> 1 + Cos[\[Theta]]}, {\[Theta], \(-\[Pi]\)/2, \[Pi]/2},
>
> Fills \[Rule] LightSteelBlue, \ Curves \[Rule] None,
> PlotPoints \[Rule] 31] /.
> DrawingTransform[Function[{\[Theta], r}, r\
> Cos[\[Theta]]],
> Function[{\[Theta], r}, r\ Sin[\[Theta]]]],
> \n\t\tDrawPolarR[
> 1 + Cos[\[Theta]], {\[Theta], 0, 2 \[Pi]},
> PlotPoints \[Rule] 31], \n\t\tDrawPolarR[
> 1, {\[Theta], 0, 2 \[Pi]}]}], \n\n
> AspectRatio \[Rule] Automatic,
> PlotRange \[Rule] {{\(-1.501\), 2.5}, {\(-2\), 2}},
> Background \[Rule] Linen, \n{Frame \[Rule] True,
> FrameLabel \[Rule] {x, y}}];\)\)], "Input"]
> }, Closed]]
> }, Open ]]
> },
> FrontEndVersion->"4.0 for Microsoft Windows",
> ScreenRectangle->{{0, 1280}, {0, 943}},
> WindowSize->{660, 763},
> WindowMargins->{{0, Automatic}, {Automatic, 0}}
> ]
>
> (***********************************************************************
> Cached data follows. If you edit this Notebook file directly, not using
> Mathematica, you must remove the line containing CacheID at the top of
> the file. The cache data will then be recreated when you save this file
>
> from within Mathematica.
> ***********************************************************************)
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> *)
>
> (***********************************************************************
> End of Mathematica Notebook file.
> ***********************************************************************)
--
Murray Eisenberg murray at math.umass.edu
Mathematics & Statistics Dept. phone 413 549-1020 (H)
Univ. of Massachusetts 413 545-2859 (W)
Amherst, MA 01003-4515
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