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Re: Just primitive ColorFunction

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87521] Re: Just primitive ColorFunction
  • From: ucervan at gmail.com
  • Date: Sat, 12 Apr 2008 06:57:58 -0400 (EDT)
  • References: <ftfej7$bu7$1@smc.vnet.net> <ftfk8g$fab$1@smc.vnet.net>

> 1.  Is there a more appropriate way to force Plot to calculate the
> function value at certain points?


How about (not very clean, but does the work if you really need to use
ColorFunction for this to make it work with Filling):

ep = 0.0001;
Plot[Sin[x], {x, 0, 4 Pi}, PlotStyle -> Thick,
 ColorFunction -> (If[Sin[#] >= 0, RGBColor[1, 0, 0],
     RGBColor[0, 0, 1]] &), ColorFunctionScaling -> False,
 Mesh -> {{Pi, Pi + ep, 2 Pi, 2 Pi + ep, 3 Pi, 3 Pi + ep}},
 MeshStyle -> None]

and if you still need to add mesh points in the x direction, use:

ep = 0.0001;
Plot[Sin[x], {x, 0, 4 Pi}, PlotStyle -> Thick,
 ColorFunction -> (If[Sin[#] >= 0, RGBColor[1, 0, 0],
     RGBColor[0, 0, 1]] &), ColorFunctionScaling -> False,
 Mesh -> {{Pi, Pi + ep, 2 Pi, 2 Pi + ep, 3 Pi, 3 Pi + ep}, 10},
 MeshStyle -> {None, Black}, MeshFunctions -> {#1 &, #1 &}]

>
> 2.  Is there a way to avoid having to find the zeros of the function
> manually?  (More generally: avoid having to calculate the points where
> the colouring changes abruptly.)

Look at these two cases:


1)

Plot[x, {x, 0, 4 Pi}, PlotStyle -> Thick,
 ColorFunction -> (If[Mod[IntegerPart[30 #], 2] == 0,
     RGBColor[1, 0, 0], RGBColor[0, 0, 1]] &)]

2)

Plot[x, {x, 0, 4 Pi}, PlotStyle -> Thick, Mesh -> 20,
 MeshShading -> {Red, Blue}]

Subdividing 1) by color will succeed only if the initial sampling gets
the colored regions right. Then a color based "find root" will need to
be computed for every single "color jump". We will also need to define
what a "jump" or color based Exclusion is.

2) is the way of dealing with subdividing curves and surfaces via the
Mesh/MeshFunctions/MeshShading options.

An Automatic intersection method can be attained in many cases (only
if the mesh functions evaluate to +/- values, so no min/max tangential
intersections) with something like:


Plot[Sin[x], {x, 0, 4 Pi}, PlotStyle -> Thick, Mesh -> {{0.}},
 MeshShading -> {Red, Blue}, MeshFunctions -> {(Sin[#]) &}]

Plot[{x, Sin[x] + x}, {x, 0, 4 Pi}, PlotStyle -> Thick,
 Mesh -> {{0.}}, MeshShading -> {Red, Blue},
 MeshFunctions -> {(Sin[#]) &}]

Plot[{Cos[x] + x, Sin[x] + x}, {x, 0, 4 Pi}, PlotStyle -> Thick,
 Mesh -> {{0.}}, MeshShading -> {Red, Blue},
 MeshFunctions -> {(Sin[#] - Cos[#]) &}]

Finally, using all ColorFunction/Mesh/MeshFunctions/Filling options
together:

Plot[{x, Sin[x] + x}, {x, 0, 4 Pi}, PlotStyle -> Thick,
 Mesh -> {{0.}}, MeshFunctions -> {(Sin[#]) &},
 ColorFunction -> (If[Sin[#] >= 0, RGBColor[1, 0, 0],
     RGBColor[0, 0, 1]] &), ColorFunctionScaling -> False,
 Filling -> {1 -> {2}}]


-Ulises Cervantes
WRI



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