Re: Problem with Min Max between two functions
- To: mathgroup at smc.vnet.net
- Subject: [mg112098] Re: Problem with Min Max between two functions
- From: "J. Batista" <jbatista800 at gmail.com>
- Date: Mon, 30 Aug 2010 06:20:21 -0400 (EDT)
Dear Maria, I reviewed your note. I did not experience the error that you
described when executing the notebook. I will directly send you the
original notebook with the additional lines of code appended to the end.
This version of your original notebook also has the graphical results I
obtained after executing it, in the form of the final image found in the
notebook.
Regards,
J. Batista
On Thu, Aug 26, 2010 at 6:49 AM, maria giovanna dainotti <
mariagiovannadainotti at yahoo.it> wrote:
> Dear J. Batista,
> I used the following equations
> R1====1.029
> R2====3.892
> R3====8
> e1====250
> e2====11.8
> e3====80.5
> i====p/12
> spherenear[x_,y_]:====((R3^2-x^2-y^2)^(1/2))
> spherefar[x_,y_]:====-((R3^2-x^2-y^2)^(1/2))
> jetnear[x_,y_]:====Min[Re[spherenear[x,y]],Re[(R1^2-x^2)^(1/2)+y*Tan[i]]]
> jetfar[x_,y_]:====Max[Re[spherefar[x,y]],Re[-(R1^2-x^2)^(1/2)]+y*Tan[i]]
> jetbrightness[x_,y_]:====Boole[-R1==EF==82==A3x==EF==82==A3R1]*(jetnear[x,y] -
> jetfar[x,y])*e1
> emptynear[x_,y_]:====Min[Re[spherenear[x,y]],Re[(R2^2-x^2)^(1/2)+y*Tan[i]]]
> emptyfar[x_,y_]:====Max[Re[spherefar[x,y]],Re[-(R2^2-x^2)^(1/2)]+y*Tan[i]]
> emptybrightness[x_,y_]:====Boole[-R1<x<R1]*(emptynear[x,y] -jetnear[x,y] +
> jetfar[x,y]- emptyfar[x,y])*e2 +(Boole[-R2
> <x<-R1]+Boole[R2>x>R1])*(emptynear[x,y] -jetnear[x,y] + jetfar[x,y]-
> emptyfar[x,y])*e2
> spherebrightness[x_,y_]:====Boole[-R2<x<R2]*(spherenear[x,y] -emptynear[x,y]
> +
> emptyfar[x,y]- spherefar[x,y])*e3
> +(Boole[x<-R2]+Boole[R2<x])*(spherenear[x,y] -
> spherefar[x,y])*e3
> ==E3==80==80
>
> f[x_,y_]:====((jetnear[x,y]-jetfar[x,y])*e1+e3*(spherenear[x,y]-emptynear[x,y]+emptyfar[x,y]-spherefar[x,y])+e2*(emptynear[x,y]-jetnear[x,y]+jetfar[x,y]-emptyfar[x,y]))*Boole[-R1<x<R1]+((emptynear[x,y]-emptyfar[x,y])*e2+e3*(spherenear[x,y]-emptynear[x,y]+emptyfar[x,y]-spherefar[x,y]))*(Boole[-R2<x<-R1]+Boole[R2>x>R1])+(spherenear[x,y]-spherefar[x,y])*e3*(Boole[x<-R2]+Boole[x>R2])
>
> ==E3==80==80
> 1.029
> 3.892
> 8
> 250
> 11.8
> 80.5p==C2
>
> originalPlot====ContourPlot[f[x,y],{x,-8,8},{y,-8,8},Contours==EF==82==AE50,ColorFunction==EF==82==AE(Hue[#]&)]/12
>
> originalColorData========ImageData[originalPlot];
>
> targetPixels========Position[originalColorData,originalColorData[[180,180=
]]];newColorData========ReplacePart[originalColorData,targetPixels
>
> ->
> Image[newColorData]
> But I got the following error message:
> Part::partd: Part specification
> \[NoBreak]originalColorData==EF==81==90180,180\[RightDoubleBracket]\[NoBr=
eak]
> is longer
> than depth of object. ==EF==82==87
> ==C2
> Image::imgarray: The specified argument
> \[NoBreak]originalColorData\[NoBreak]
> should be an array of rank 2 or 3 with machine-size numbers. ==EF==82==87
> I will be very grateful if you could help me.
> Thanks a lot
> Best regards
> Maria
> ==C2 {1.,1.,1.}];
>
> ________________________________
> Da: J. Batista <jbatista800 at gmail.com>
> A: mathgroup at smc.vnet.net
> Inviato: Lun 2 agosto 2010, 13:02:00
> Oggetto: [mg111445] Re: Problem with Min Max between two functions
>
> Maria/All, I just learned from a colleague that all equations on my messa=
ge
> have double equal signs.==C2 Please note that is probably due to a trans=
m==
> ission
> error.==C2 All equations should only have a single equal sign.==C2 I'm
> retransmitting my original message.
> Regards,
> J. Batista
> On Sun, Aug 1, 2010 at 3:10 AM, J. Batista <jbatista800 at gmail.com> wrote:
>
> > Dean Maria, here is a possible solution to your question.==C2 The whit=
e
> > contour areas can be removed by approaching the output of ContourPlot a=
s
> ==
> an
> > image, treating the task as one of image processing (as suggested
> previou==
> sly
> > by Daniel Lichtblau).==C2 First, equate the original ContourPlot outpu=
t==
> with a
> > variable name, for example originalPlot ======== Out[1]==C2 (where Out=
[1] i==
> s the
> > ContourPlot output cell).==C2 Alternatively, you can simply select the
> > ContourPlot, copy and then paste into the first line of the code sequen=
ce
> > below in place of the variable originalPlot.==C2 I will now display th=
e==
> four
> > lines of the code sequence and then explain them afterwards.
> >
> > originalColorData ======== ImageData[originalPlot];
> >
> > targetPixels ======== Position[originalColorData, originalColorData[[18=
0,
> 180==
> ]]];
> >
> > newColorData ======== ReplacePart[originalColorData, targetPixels -> {1=
., 1.,
> > 1.}];
> >
> > Image[newColorData]
> >
> >
> > The first code line accomplishes the task of collecting and reading the
> > ContourPlot into computer memory as image data, in this case a vector o=
f
> ==
> RGB
> > color values.==C2 Note that I place a semicolon at the end of this and=
==
> other
> > lines of code in order to suppress the visual output of the code
> sequence==
> 's
> > result.==C2 This is because the vector is lengthy and will clog the no=
t==
> ebook
> > unnecessarily.
> > The second code line establishes the pattern by which the portions of t=
he
> > plot that you wish to alter are identified as a subset of the entire
> > original data set.==C2 The pattern is established by entering the pixe=
l
> > coordinate of a representative target pixel that you wish to alter, in
> th==
> is
> > case [[180, 180]] being one of the pixels in the white contour areas.===
C2==
> You
> > can determine an appropriate pixel coordinate by right-clicking in the
> > original ContourPlot output, selecting Get Indices, and then guiding yo=
ur
> > cursor to a desired location within the plot.
> > The third code line replaces the pixel locations flagged by the previou=
s
> > pattern search with new pixel data, in this case new RGB color values
> tha==
> t
> > you select.==C2 I have used the example of {1., 1., 1.} to illustrate =
c==
> hanging
> > from the semi-white color of the original plot to a true white that
> match==
> es
> > the plot background.==C2 Be sure to use decimal points as above when e=
x==
> pressing
> > color values for your pixels, as something like {1, 1, 1} will not
> > be understood correctly for this purpose.==C2 If you want to change th=
e
> > semi-white color of your original plot to black, use {0., 0., 0.}.
> > The fourth and final line re-establishes the newly altered set of pixel
> > data as an image object, and displays the altered image.
> >
> > Hope this helps.
> > Best Regards,
> > J. Batista
> >
> >==C2 On Mon, Jul 26, 2010 at 6:37 AM, maria giovanna dainotti <
> > mariagiovannadainotti at yahoo.it> wrote:
> >
> >> Dear Mathgroup,
> >> I have the following function
> >> R1========1.029
> >> R2========3.892
> >> R3========8
> >> e1========250
> >> e2========11.8
> >> e3========80.5
> >> i========pi/12
> >> spherenear[x_,y_]:========((R3^2-x^2-y^2)^(1/2))
> >> spherefar[x_,y_]:========-((R3^2-x^2-y^2)^(1/2))
> >>
> emptynear[x_,y_]:========Min[Re[spherenear[x,y]],Re[(R2^2-x^2)^(1/2)+y*Ta=
n[i==
> ]]====
> ]
> >>
> emptyfar[x_,y_]:========Max[Re[spherefar[x,y]],Re[-(R2^2-x^2)^(1/2)]+y*Ta=
n[i==
> ]]
> >>
> jetnear[x_,y_]:========Min[Re[spherenear[x,y]],Re[(R1^2-x^2)^(1/2)+y*Tan[=
i]]==
> ]
> >> jetfar[x_,y_]:========Max[Re[spherefar[x,y]],Re[-(R1^2-x^2)^(1/2)]+y*T=
an[i]]
> >>
> >>
> f[x_,y_]:========((jetnear[x,y]-jetfar[x,y])*e1+e3*(spherenear[x,y]-empty=
nea==
> r[====
>
> x,y]+emptyfar[x,y]-spherefar[x,y])+e2*(emptynear[x,y]-jetnear[x,y]+jetfar=
[x==
> ====
>
> ,y]-emptyfar[x,y]))*Boole[-R1========EF========82========A3x========EF===
======82========A3R1]+((emp==
> tynear[x,y]-====
>
> emptyfar[x,y])*e2+e3*(spherenear[x,y]-emptynear[x,y]+emptyfar[x,y]-sphere=
fa==
> ====
>
> r[x,y]))*(Boole[-R2<x<-R1]+Boole[R2>x>R1])+(spherenear[x,y]-spherefar[x,y=
])==
> ====
> *e3*(Boole[x<-R2]+Boole[x>R2])
> >>
> >>
> >>
> ContourPlot[f[x,y],{x,-8,8},{y,-8,8},Contours========EF========82=========
AE{0,100,20==
> 0,300,====
> 400,500,600,700,800,900}]
> >>
> >> From the picture you can see there is a white contours that results a
> bi==
> ====
> t
> >> odd, I
> >> think th at it comes out from the introduction of the Min and Max.
> >> I would like to remove this white contour. Could you help me?
> >> Thanks a lot for your attention
> >> Cheers
> >> Maria
> >>
> >>
> >
>
>
>
>
> --0-882231370-1282758216==:97664
> Content-Type: text/html; charset=="utf-8"
> Content-Transfer-Encoding: quoted-printable
> X-Sun-Content-Length: 10136
>
> <html><head><style type=="text/css"><!-- DIV {margin:0px;} --></style></h=
e==
> ad><body><div style=="font-family:arial, helvetica, sans-serif;font-size:=
1==
> 4pt"><DIV>Dear J. Batista,</DIV>==0A<DIV>I used the following
> equations</DIV==
> >==0A<DIV>==0A<P>R1==1.029</P>==0A<P>R2==3.892</P>==0A<P>R3==8</P>==0A<P>=
e1==
> ==250</P>==0A<P>e2==11.8</P>==0A<P>e3==80.5</P>==0A<P>i==<FONT face==Math=
==
> ematica1Mono>p</FONT>/12</P>==0A<P>spherenear[x_,y_]:==((R3^2-x^2-y^2)^(1=
/2==
> ))</P>==0A<P>spherefar[x_,y_]:==-((R3^2-x^2-y^2)^(1/2))</P>==0A<P>jetnear=
[x_==
> ,y_]:==Min[Re[spherenear[x,y]],Re[(R1^2-x^2)^(1/2)+y*Tan[i]]]</P>==0A<P>j=
et==
> far[x_,y_]:==Max[Re[spherefar[x,y]],Re[-(R1^2-x^2)^(1/2)]+y*Tan[i]]</P>===
0A==
> <P>jetbrightness[x_,y_]:==Boole[-R1<FONT face==Mathematica1Mono>==EF==82=
==A3==
> </FONT>x<FONT face==Mathematica1Mono>==EF==82==A3</FONT>R1]*(jetnear[x,y]=
- j==
> etfar[x,y])*e1</P>==0A<P>emptynear[x_,y_]:==Min[Re[spherenear[x,y]],Re[(R=
2^==
> 2-x^2)^(1/2)+y*Tan[i]]]</P>==0A<P>emptyfar[x_,y_]:==Max[Re[spherefar[x,y]=
],==
> Re[-(R2^2-x^2)^(1/2)]+y*Tan[i]]</P>==0A<P>emptybrightness[x_,y_]:==Boole[=
-R==
> 1<FONT face==Mathematica1Mono><</FONT>x<FONT face==Mathematica1Mono>&l=
==
> t;</FONT>R1]*(emptynear[x,y] -jetnear[x,y] + jetfar[x,y]-
> emptyfar[x,y])*e2==
> +(Boole[-R2 <x<-R1]+Boole[R2>x>R1])*(emptynear[x,y]
> -jetnear[x==
> ,y] + jetfar[x,y]- emptyfar[x,y])*e2</P>==0A<P>spherebrightness[x_,y_]:===
Bo==
> ole[-R2<FONT face==Mathematica1Mono><</FONT>x<FONT face==Mathematica1M=
==
> ono><</FONT>R2]*(spherenear[x,y] -emptynear[x,y] + emptyfar[x,y]-
> sphere==
> far[x,y])*e3 +(Boole[x<-R2]+Boole[R2<x])*(spherenear[x,y] -
> spherefar==
> [x,y])*e3</P>==0A<P>==E3==80==80</P>==0A<P>f[x_,y_]:==((jetnear[x,y]-jetf=
ar[x,y==
>
> ])*e1+e3*(spherenear[x,y]-emptynear[x,y]+emptyfar[x,y]-spherefar[x,y])+e2=
*(==
> emptynear[x,y]-jetnear[x,y]+jetfar[x,y]-emptyfar[x,y]))*Boole[-R1<FONT
> face==
> ==Mathematica1Mono><</FONT>x<FONT face==Mathematica1Mono><</FONT>R1=
==
>
> ]+((emptynear[x,y]-emptyfar[x,y])*e2+e3*(spherenear[x,y]-emptynear[x,y]+e=
mp==
>
> tyfar[x,y]-spherefar[x,y]))*(Boole[-R2<x<-R1]+Boole[R2>x>R1])=
+(==
>
> spherenear[x,y]-spherefar[x,y])*e3*(Boole[x<-R2]+Boole[x>R2])</P>===
0A<==
>
> P>==E3==80==80</P>==0A<P>1.029</P>==0A<P>3.892</P>==0A<P>8</P>==0A<P>250<=
/P>==0A<P>==
> 11.8</P>==0A<P>80.5</P><FONT face==Mathematica1Mono>==0A<P>p</FONT><FONT =
fac==
> e=="Times New Roman">/12</FONT></P> <BR><B><FONT face==Courier>==0A<=
P==
> >originalPlot==ContourPlot[f[x,y],{x,-8,8},{y,-8,8},Contours</FONT><FONT =
f==
> ace==Mathematica1Mono>==EF==82==AE</FONT><FONT face==Courier>50,ColorFunc=
tio==
> n</FONT><FONT face==Mathematica1Mono>==EF==82==AE</FONT><FONT face==Couri=
er>==
> (Hue[#]&)]</P></B></FONT></DIV>==0A<DIV style=="FONT-FAMILY: arial, h=
el==
> vetica, sans-serif; FONT-SIZE: 14pt"><B><FONT face==Courier>==0A<P align=
====
> left>originalColorData====ImageData[originalPlot];</P>==0A<P align==left>=
==
> targetPixels====Position[originalColorData,originalColorData[[180,180]]];=
==
> newColorData====ReplacePart[originalColorData,targetPixels</P>==0A<P alig=
n==
> ==left>-></FONT><FONT face==Courier>{1.,1.,1.}];</P>==0A<P>Image[newCo=
l==
> orData]</P>==0A<P>But I got the following error message:
> </P>==0A<P>Part::par==
> td: Part specification \[NoBreak]originalColorData<FONT face==Mathematica=
2==
> >==EF==81==90</FONT>180,180\[RightDoubleBracket]\[NoBreak] is longer than
> dept==
> h of object. <A href=="http://reference.wolfram.com/mathematica/ref/Part.=
h==
> tml"><FONT face==Mathematica1>==EF==82==87</FONT></A></P>==0A<P> </P=
>==0A<==
> P>Image::imgarray: The specified argument
> \[NoBreak]originalColorData\[NoBr==
> eak] should be an array of rank 2 or 3 with machine-size numbers. <A href=
==
> =="http://reference.wolfram.com/mathematica/ref/Image.html";><FONT face==M=
==
> athematica1>==EF==82==87</FONT></A></P>==0A<P>I will be very grateful if =
you
> co==
> uld help me.</P>==0A<P>Thanks a lot</P>==0A<P>Best
> regards</P>==0A<P>Maria</P>==
> ==0A<P> </P></B></FONT>==0A<DIV style=="FONT-FAMILY: arial, helvetic=
a, ==
> sans-serif; FONT-SIZE: 13px"><FONT size==2 face==Tahoma>==0A<HR SIZE==1>=
==
> ==0A<B><SPAN style=="FONT-WEIGHT: bold">Da:</SPAN></B> J. Batista <jba=
ti==
> sta800 at gmail.com><BR><B><SPAN style=="FONT-WEIGHT: bold">A:</SPAN></B>=
==
> mathgroup at smc.vnet.net<BR><B><SPAN style=="FONT-WEIGHT: bold">Inviato:</S=
P==
> AN></B> Lun 2 agosto 2010, 13:02:00<BR><B><SPAN style=="FONT-WEIGHT: bold=
"==
> >Oggetto:</SPAN></B> [mg111445] Re: Problem with Min Max between two
> functi==
> ons<BR></FONT><BR>Maria/All, I just learned from a colleague that all
> equat==
> ions on my message<BR>have double equal signs. Please note that is
> pr==
> obably due to a transmission<BR>error. All equations should only
> have==
> a single equal sign. I'm<BR>retransmitting my original
> message.<BR>R==
> egards,<BR>J. Batista<BR>On Sun, Aug 1, 2010 at 3:10 AM, J. Batista <<=
A
> ==
> href=="mailto:jbatista800 at gmail.com" ymailto=="mailto:jbatista800 at gmail.c=
==
> om">jbatista800 at gmail.com</A>> wrote:<BR><BR>> Dean Maria, here is =
a
> ==
> possible solution to your question. The white<BR>> contour areas
> c==
> an be
> removed by approaching the output of ContourPlot as an<BR>> image,
> trea==
> ting the task as one of image processing (as suggested previously<BR>>
> b==
> y Daniel Lichtblau). First, equate the original ContourPlot output
> wi==
> th a<BR>> variable name, for example originalPlot ==== Out[1] (w=
==
> here Out[1] is the<BR>> ContourPlot output cell). Alternatively,
> y==
> ou can simply select the<BR>> ContourPlot, copy and then paste into th=
e
> ==
> first line of the code sequence<BR>> below in place of the variable
> orig==
> inalPlot. I will now display the four<BR>> lines of the code
> seque==
> nce and then explain them afterwards.<BR>><BR>> originalColorData =
====
> == ImageData[originalPlot];<BR>><BR>> targetPixels ==== Position[o=
==
> riginalColorData, originalColorData[[180, 180]]];<BR>><BR>>
> newColorD==
> ata ==== ReplacePart[originalColorData, targetPixels -> {1., 1.,<BR>&g=
==
> t; 1.}];<BR>><BR>> Image[newColorData]<BR>><BR>><BR>> The
> fi==
> rst code
> line accomplishes the task of collecting and reading the<BR>>
> ContourPl==
> ot into computer memory as image data, in this case a vector of
> RGB<BR>>==
> color values. Note that I place a semicolon at the end of this and
> o==
> ther<BR>> lines of code in order to suppress the visual output of the
> co==
> de sequence's<BR>> result. This is because the vector is lengthy
> a==
> nd will clog the notebook<BR>> unnecessarily.<BR>> The second code
> li==
> ne establishes the pattern by which the portions of the<BR>> plot that
> y==
> ou wish to alter are identified as a subset of the entire<BR>> origina=
l
> ==
> data set. The pattern is established by entering the pixel<BR>>
> co==
> ordinate of a representative target pixel that you wish to alter, in
> this<B==
> R>> case [[180, 180]] being one of the pixels in the white contour
> areas==
> . You<BR>> can determine an appropriate pixel coordinate by
> right-==
> clicking in the<BR>> original ContourPlot output, selecting Get
> Indices, and then guiding your<BR>> cursor to a desired location
> within==
> the plot.<BR>> The third code line replaces the pixel locations
> flagged==
> by the previous<BR>> pattern search with new pixel data, in this case
> n==
> ew RGB color values that<BR>> you select. I have used the exampl=
e
> ==
> of {1., 1., 1.} to illustrate changing<BR>> from the semi-white color
> of==
> the original plot to a true white that matches<BR>> the plot
> background==
> . Be sure to use decimal points as above when expressing<BR>>
> colo==
> r values for your pixels, as something like {1, 1, 1} will not<BR>> be
> u==
> nderstood correctly for this purpose. If you want to change
> the<BR>&g==
> t; semi-white color of your original plot to black, use {0., 0.,
> 0.}.<BR>&g==
> t; The fourth and final line re-establishes the newly altered set of
> pixel<==
> BR>> data as an image object, and displays the altered
> image.<BR>><BR==
> >> Hope this helps.<BR>> Best Regards,<BR>> J.
> Batista<BR>><BR>> On Mon, Jul 26, 2010 at 6:37 AM, maria
> giova==
> nna dainotti <<BR>> <A href=="mailto:mariagiovannadainotti at yahoo.it=
"==
> ymailto=="mailto:mariagiovannadainotti at yahoo.it">mariagiovannadainotti@y=
a==
> hoo.it</A>> wrote:<BR>><BR>>> Dear Mathgroup,<BR>>> I
> hav==
> e the following function<BR>>> R1====1.029<BR>>> R2====3.89==
> 2<BR>>> R3====8<BR>>> e1====250<BR>>> e2====11.8<BR==
> >>> e3====80.5<BR>>> i====pi/12<BR>>> spherenear[x_,y==
> _]:====((R3^2-x^2-y^2)^(1/2))<BR>>> spherefar[x_,y_]:====-((R3^2-==
> x^2-y^2)^(1/2))<BR>>> emptynear[x_,y_]:====Min[Re[spherenear[x,y]],=
==
> Re[(R2^2-x^2)^(1/2)+y*Tan[i]]==<BR>]<BR>>> emptyfar[x_,y_]:====Max=
==
> [Re[spherefar[x,y]],Re[-(R2^2-x^2)^(1/2)]+y*Tan[i]]<BR>>>
> jetnear[x_,==
> y_]:====Min[Re[spherenear[x,y]],Re[(R1^2-x^2)^(1/2)+y*Tan[i]]]<BR>>>=
==
> ; jetfar[x_,y_]:====Max[Re[spherefar[x,y]],Re[-(R1^2-x^2)^(1/2)]+y*Tan[i]=
==
> ]<BR>>><BR>>>
> f[x_,y_]:====((jetnear[x,y]-jetfar[x,y])*e1+e3*(spherenear[x,y]-emptynea=
==
> r[==<BR>x,y]+emptyfar[x,y]-spherefar[x,y])+e2*(emptynear[x,y]-jetnear[x,y=
]==
> +jetfar[x==<BR>,y]-emptyfar[x,y]))*Boole[-R1====EF====82====A3x====
> ==EF====82====A3R1]+((emptynear[x,y]-==<BR>emptyfar[x,y])*e2+e3*(sphe==
> renear[x,y]-emptynear[x,y]+emptyfar[x,y]-spherefa==<BR>r[x,y]))*(Boole[-R=
2==
> <x<-R1]+Boole[R2>x>R1])+(spherenear[x,y]-spherefar[x,y])==<BR=
>==
> *e3*(Boole[x<-R2]+Boole[x>R2])<BR>>><BR>>><BR>>>
> Co==
> ntourPlot[f[x,y],{x,-8,8},{y,-8,8},Contours====EF====82====AE{0,100,2==
> 00,300,==<BR>400,500,600,700,800,900}]<BR>>><BR>>> From the p=
i==
> cture you can see there is a white contours that results a bi==<BR>t<BR>&=
g==
> t;> odd, I<BR>>> think th at it comes out from the introduction
> of==
> the Min and Max.<BR>>> I would like to remove this white contour.
> Co==
> uld you help me?<BR>>> Thanks a lot for your attention<BR>>>
> Ch==
> eers<BR>>>
>
> Maria<BR>>><BR>>><BR>><BR><BR><BR></DIV></DIV></div><br>=
==0A==
> ==0A==0A==0A </body></html>
> --0-882231370-1282758216==:97664--
>
>
--00163691fc66c5736f048efc72a6
Content-Type: text/html; charset="ISO-8859-1"
Content-Transfer-Encoding: quoted-printable
X-Sun-Content-Length: 24381
<div>Dear Maria, I reviewed your note. I did not experience the error th=
at you described when executing the notebook. I will directly send you t=
he original notebook with the additional lines of code appended to the end.=
This version of your original notebook also has the graphical results I=
obtained after executing it, in the form of the final image found in the n=
otebook.</div>
<div>Regards,</div>
<div>J. Batista<br><br></div>
<div class="gmail_quote">On Thu, Aug 26, 2010 at 6:49 AM, maria giovanna =
dainotti <span dir="ltr"><<a href="mailto:mariagiovannadainotti@yaho=
o.it">mariagiovannadainotti at yahoo.it</a>></span> wrote:<br>
<blockquote style="BORDER-LEFT: #ccc 1px solid; MARGIN: 0px 0px 0px 0.8ex=
; PADDING-LEFT: 1ex" class="gmail_quote">Dear J. Batista,<br>I used the f=
ollowing equations<br>R1==1.029<br>R2==3.892<br>R3==8<br>e1==
=250<br>e2==11.8<br>
e3==80.5<br>i==p/12<br>spherenear[x_,y_]:==((R3^2-x^2-y^2)^(1/2=
))<br>spherefar[x_,y_]:==-((R3^2-x^2-y^2)^(1/2))<br>jetnear[x_,y_]:==
=Min[Re[spherenear[x,y]],Re[(R1^2-x^2)^(1/2)+y*Tan[i]]]<br>jetfar[x_,y_]:=
==Max[Re[spherefar[x,y]],Re[-(R1^2-x^2)^(1/2)]+y*Tan[i]]<br>
jetbrightness[x_,y_]:==Boole[-R1=EF=82=A3x=EF=82=A3R1]*(jet=
near[x,y] - jetfar[x,y])*e1<br>emptynear[x_,y_]:==Min[Re[spherenear[x,y=
]],Re[(R2^2-x^2)^(1/2)+y*Tan[i]]]<br>emptyfar[x_,y_]:==Max[Re[spherefar=
[x,y]],Re[-(R2^2-x^2)^(1/2)]+y*Tan[i]]<br>
emptybrightness[x_,y_]:==Boole[-R1<x<R1]*(emptynear[x,y] -jetnear=
[x,y] +<br>jetfar[x,y]- emptyfar[x,y])*e2 +(Boole[-R2<br><x<-R1]+Bool=
e[R2>x>R1])*(emptynear[x,y] -jetnear[x,y] + jetfar[x,y]-<br>emptyfar[=
x,y])*e2<br>
spherebrightness[x_,y_]:==Boole[-R2<x<R2]*(spherenear[x,y] -empty=
near[x,y] +<br>emptyfar[x,y]- spherefar[x,y])*e3 +(Boole[x<-R2]+Boole[R2=
<x])*(spherenear[x,y] -<br>spherefar[x,y])*e3<br>=E3=80=80<br>f[x_=
,y_]:==((jetnear[x,y]-jetfar[x,y])*e1+e3*(spherenear[x,y]-emptynear[x,y=
]+emptyfar[x,y]-spherefar[x,y])+e2*(emptynear[x,y]-jetnear[x,y]+jetfar[x,y]=
-emptyfar[x,y]))*Boole[-R1<x<R1]+((emptynear[x,y]-emptyfar[x,y])*e2+e=
3*(spherenear[x,y]-emptynear[x,y]+emptyfar[x,y]-spherefar[x,y]))*(Boole[-R2=
<x<-R1]+Boole[R2>x>R1])+(spherenear[x,y]-spherefar[x,y])*e3*(Bo=
ole[x<-R2]+Boole[x>R2])<br>
<br>=E3=80=80<br>1.029<br>3.892<br>8<br>250<br>11.8<br>80.5p=C2<br>=
originalPlot==ContourPlot[f[x,y],{x,-8,8},{y,-8,8},Contours=EF=82=
=AE50,ColorFunction=EF=82=AE(Hue[#]&)]/12<br><br>originalColorD=
ata====ImageData[originalPlot];<br>
targetPixels====Position[originalColorData,originalColorData[[180,1=
80]]];newColorData====ReplacePart[originalColorData,targetPixels<br=
><br>-><br>Image[newColorData]<br>But I got the following error message:=
<br>Part::partd: Part specification<br>
\[NoBreak]originalColorData=EF=81=90180,180\[RightDoubleBracket]\[NoB=
reak] is longer<br>than depth of object. =EF=82=87<br>=C2<br>Image:=
:imgarray: The specified argument \[NoBreak]originalColorData\[NoBreak]<br>=
should be an array of rank 2 or 3 with machine-size numbers. =EF=82=8=
7<br>
I will be very grateful if you could help me.<br>Thanks a lot<br>Best regar=
ds<br>Maria<br>=C2 {1.,1.,1.}];<br><br>________________________________<b=
r>Da: J. Batista <<a href="mailto:jbatista800 at gmail.com">jbatista800@g=
mail.com</a>><br>
A: <a href="mailto:mathgroup at smc.vnet.net">mathgroup at smc.vnet.net</a><br>=
Inviato: Lun 2 agosto 2010, 13:02:00<br>Oggetto: [mg111445] Re: Problem wit=
h Min Max between two functions<br><br>Maria/All, I just learned from a col=
league that all equations on my message<br>
have double equal signs.=C2 Please note that is probably due to a tran=
sm=<br>ission<br>error.=C2 All equations should only have a single e=
qual sign.=C2 I'm<br>retransmitting my original message.<br>Regard=
s,<br>J. Batista<br>
On Sun, Aug 1, 2010 at 3:10 AM, J. Batista <<a href="mailto:jbatista80=
0 at gmail.com">jbatista800 at gmail.com</a>> wrote:<br><br>> Dean Maria, h=
ere is a possible solution to your question.=C2 The white<br>> cont=
our areas can be removed by approaching the output of ContourPlot as =<br=
>
an<br>> image, treating the task as one of image processing (as suggeste=
d previou=<br>sly<br>> by Daniel Lichtblau).=C2 First, equate the=
original ContourPlot output=<br> with a<br>> variable name, for exa=
mple originalPlot ==== Out[1]=C2 (where Out[1] i=<br>
s the<br>> ContourPlot output cell).=C2 Alternatively, you can simp=
ly select the<br>> ContourPlot, copy and then paste into the first line =
of the code sequence<br>> below in place of the variable originalPlot.=
=C2 I will now display the=<br>
four<br>> lines of the code sequence and then explain them afterwards=
.<br>><br>> originalColorData ==== ImageData[originalPlot];<b=
r>><br>> targetPixels ==== Position[originalColorData, origin=
alColorData[[180, 180=<br>
]]];<br>><br>> newColorData ==== ReplacePart[originalColorDat=
a, targetPixels -> {1., 1.,<br>> 1.}];<br>><br>> Image[newColor=
Data]<br>><br>><br>> The first code line accomplishes the task of =
collecting and reading the<br>
> ContourPlot into computer memory as image data, in this case a vector =
of =<br>RGB<br>> color values.=C2 Note that I place a semicolon a=
t the end of this and =<br>other<br>> lines of code in order to suppre=
ss the visual output of the code sequence=<br>
's<br>> result.=C2 This is because the vector is lengthy and wi=
ll clog the not=<br>ebook<br>> unnecessarily.<br>> The second code =
line establishes the pattern by which the portions of the<br>> plot that=
you wish to alter are identified as a subset of the entire<br>
> original data set.=C2 The pattern is established by entering the =
pixel<br>> coordinate of a representative target pixel that you wish to =
alter, in th=<br>is<br>> case [[180, 180]] being one of the pixels in =
the white contour areas.=C2=<br>
You<br>> can determine an appropriate pixel coordinate by right-click=
ing in the<br>> original ContourPlot output, selecting Get Indices, and =
then guiding your<br>> cursor to a desired location within the plot.<br>
> The third code line replaces the pixel locations flagged by the previo=
us<br>> pattern search with new pixel data, in this case new RGB color v=
alues tha=<br>t<br>> you select.=C2 I have used the example of {1=
., 1., 1.} to illustrate c=<br>
hanging<br>> from the semi-white color of the original plot to a true wh=
ite that match=<br>es<br>> the plot background.=C2 Be sure to use=
decimal points as above when ex=<br>pressing<br>> color values for yo=
ur pixels, as something like {1, 1, 1} will not<br>
> be understood correctly for this purpose.=C2 If you want to chang=
e the<br>> semi-white color of your original plot to black, use {0., 0.,=
0.}.<br>> The fourth and final line re-establishes the newly altered se=
t of pixel<br>
> data as an image object, and displays the altered image.<br>><br>&g=
t; Hope this helps.<br>> Best Regards,<br>> J. Batista<br>><br>>=
;=C2 On Mon, Jul 26, 2010 at 6:37 AM, maria giovanna dainotti <<br>
> <a href="mailto:mariagiovannadainotti at yahoo.it">mariagiovannadainott=
i at yahoo.it</a>> wrote:<br>><br>>> Dear Mathgroup,<br>>> I=
have the following function<br>>> R1====1.029<br>>> R2=
====3.892<br>
>> R3====8<br>>> e1====250<br>>> e2===
==11.8<br>>> e3====80.5<br>>> i====pi/12<br=
>>> spherenear[x_,y_]:====((R3^2-x^2-y^2)^(1/2))<br>>> =
spherefar[x_,y_]:====-((R3^2-x^2-y^2)^(1/2))<br>
>> emptynear[x_,y_]:====Min[Re[spherenear[x,y]],Re[(R2^2-x^2)=
^(1/2)+y*Tan[i=<br>]]==<br>]<br>>> emptyfar[x_,y_]:=====
Max[Re[spherefar[x,y]],Re[-(R2^2-x^2)^(1/2)]+y*Tan[i=<br>]]<br>>> j=
etnear[x_,y_]:====Min[Re[spherenear[x,y]],Re[(R1^2-x^2)^(1/2)+y*Tan=
[i]]=<br>
]<br>>> jetfar[x_,y_]:====Max[Re[spherefar[x,y]],Re[-(R1^2-x^=
2)^(1/2)]+y*Tan[i]]<br>>><br>>> f[x_,y_]:====((jetnear[=
x,y]-jetfar[x,y])*e1+e3*(spherenear[x,y]-emptynea=<br>r[==<br>x,y]+em=
ptyfar[x,y]-spherefar[x,y])+e2*(emptynear[x,y]-jetnear[x,y]+jetfar[x=<br>
==<br>,y]-emptyfar[x,y]))*Boole[-R1====EF====82===
==A3x====EF====82====A3R1]+((emp=<br>tynear[x=
,y]-==<br>emptyfar[x,y])*e2+e3*(spherenear[x,y]-emptynear[x,y]+emptyfar=
[x,y]-spherefa=<br>==<br>r[x,y]))*(Boole[-R2<x<-R1]+Boole[R2>=
;x>R1])+(spherenear[x,y]-spherefar[x,y])=<br>
==<br>*e3*(Boole[x<-R2]+Boole[x>R2])<br>>><br>>><br>&=
gt;> ContourPlot[f[x,y],{x,-8,8},{y,-8,8},Contours====EF===
==82====AE{0,100,20=<br>0,300,==<br>400,500,600,700,800,9=
00}]<br>>><br>>> From the picture you can see there is a white =
contours that results a bi=<br>
==<br>t<br>>> odd, I<br>>> think th at it comes out from th=
e introduction of the Min and Max.<br>>> I would like to remove this =
white contour. Could you help me?<br>>> Thanks a lot for your attenti=
on<br>
>> Cheers<br>>> Maria<br>>><br>>><br>><br><br><b=
r><br><br>--0-882231370-1282758216=:97664<br>Content-Type: text/html; cha=
rset="utf-8"<br>Content-Transfer-Encoding: quoted-printable<br>
X-Sun-Content-Length: 10136<br><br><html><head><style type=
="text/css"><!-- DIV {margin:0px;} --></style>&l=
t;/he=<br>ad><body><div style="font-family:arial, helv=
etica, sans-serif;font-size:1=<br>
4pt"><DIV>Dear J. Batista,</DIV>=0A<DIV>I used =
the following equations</DIV=<br>>=0A<DIV>=0A<P>R1=
=1.029</P>=0A<P>R2=3.892</P>=0A<P>R3=8<=
;/P>=0A<P>e1=<br>
=250</P>=0A<P>e2=11.8</P>=0A<P>e3=80.5<=
;/P>=0A<P>i=<FONT face=Math=<br>ematica1Mono>p</F=
ONT>/12</P>=0A<P>spherenear[x_,y_]:=((R3^2-x^2-y^2)^(1/2=
=<br>))</P>=0A<P>spherefar[x_,y_]:=-((R3^2-x^2-y^2)^(1/2)=
)</P>=0A<P>jetnear[x_=<br>
,y_]:=Min[Re[spherenear[x,y]],Re[(R1^2-x^2)^(1/2)+y*Tan[i]]]</P>==
0A<P>jet=<br>far[x_,y_]:=Max[Re[spherefar[x,y]],Re[-(R1^2-x^2)^(1=
/2)]+y*Tan[i]]</P>=0A=<br><P>jetbrightness[x_,y_]:=Boole[=
-R1<FONT face=Mathematica1Mono>=EF=82=A3=<br>
</FONT>x<FONT face=Mathematica1Mono>=EF=82=A3</FONT&=
gt;R1]*(jetnear[x,y] - j=<br>etfar[x,y])*e1</P>=0A<P>emptyn=
ear[x_,y_]:=Min[Re[spherenear[x,y]],Re[(R2^=<br>2-x^2)^(1/2)+y*Tan[i]]]=
</P>=0A<P>emptyfar[x_,y_]:=Max[Re[spherefar[x,y]],=<br>
Re[-(R2^2-x^2)^(1/2)]+y*Tan[i]]</P>=0A<P>emptybrightness[x_,y=
_]:=Boole[-R=<br>1<FONT face=Mathematica1Mono>&lt;</FONT=
>x<FONT face=Mathematica1Mono>&l=<br>t;</FONT>R1]*(e=
mptynear[x,y] -jetnear[x,y] + jetfar[x,y]- emptyfar[x,y])*e2=<br>
+(Boole[-R2 &lt;x&lt;-R1]+Boole[R2&gt;x&gt;R1])*(emptyne=
ar[x,y] -jetnear[x=<br>,y] + jetfar[x,y]- emptyfar[x,y])*e2</P>=0=
A<P>spherebrightness[x_,y_]:=Bo=<br>ole[-R2<FONT face=Mathem=
atica1Mono>&lt;</FONT>x<FONT face=Mathematica1M=<br>
ono>&lt;</FONT>R2]*(spherenear[x,y] -emptynear[x,y] + emptyfar=
[x,y]- sphere=<br>far[x,y])*e3 +(Boole[x&lt;-R2]+Boole[R2&lt;x])*=
(spherenear[x,y] - spherefar=<br>[x,y])*e3</P>=0A<P>=E3=
=80=80</P>=0A<P>f[x_,y_]:=((jetnear[x,y]-jetfar[x,y=<=
br>
])*e1+e3*(spherenear[x,y]-emptynear[x,y]+emptyfar[x,y]-spherefar[x,y])+e2*(=
=<br>emptynear[x,y]-jetnear[x,y]+jetfar[x,y]-emptyfar[x,y]))*Boole[-R1<=
;FONT face=<br>=Mathematica1Mono>&lt;</FONT>x<FONT face=
=Mathematica1Mono>&lt;</FONT>R1=<br>
]+((emptynear[x,y]-emptyfar[x,y])*e2+e3*(spherenear[x,y]-emptynear[x,y]+emp=
=<br>tyfar[x,y]-spherefar[x,y]))*(Boole[-R2&lt;x&lt;-R1]+Boole[R2=
&gt;x&gt;R1])+(=<br>spherenear[x,y]-spherefar[x,y])*e3*(Boole[x&a=
mp;lt;-R2]+Boole[x&gt;R2])</P>=0A<=<br>
P>=E3=80=80</P>=0A<P>1.029</P>=0A<P>3.=
892</P>=0A<P>8</P>=0A<P>250</P>=0A<P=
>=<br>11.8</P>=0A<P>80.5</P><FONT face=Mathem=
atica1Mono>=0A<P>p</FONT><FONT fac=<br>
e="Times New Roman">/12</FONT></P>&nbsp;<=
BR><B><FONT face=Courier>=0A<P=<br>>originalPlot=
=ContourPlot[f[x,y],{x,-8,8},{y,-8,8},Contours</FONT><FONT f=<=
br>ace=Mathematica1Mono>=EF=82=AE</FONT><FONT face=Co=
urier>50,ColorFunctio=<br>
n</FONT><FONT face=Mathematica1Mono>=EF=82=AE</FONT&=
gt;<FONT face=Courier>=<br>(Hue[#]&amp;)]</P></B>=
</FONT></DIV>=0A<DIV style="FONT-FAMILY: arial, hel=
=<br>vetica, sans-serif; FONT-SIZE: 14pt"><B><FONT face=
=Courier>=0A<P align==<br>
left>originalColorData==ImageData[originalPlot];</P>=0A<P=
align=left>=<br>targetPixels==Position[originalColorData,origin=
alColorData[[180,180]]];=<br>newColorData==ReplacePart[originalColorD=
ata,targetPixels</P>=0A<P align=<br>
=left>-&gt;</FONT><FONT face=Courier>{1.,1.,1.}];<=
;/P>=0A<P>Image[newCol=<br>orData]</P>=0A<P>But =
I got the following error message: </P>=0A<P>Part::par=<br>=
td: Part specification \[NoBreak]originalColorData<FONT face=Mathemati=
ca2=<br>
>=EF=81=90</FONT>180,180\[RightDoubleBracket]\[NoBreak] is l=
onger than dept=<br>h of object. <A href="<a href="http://ref=
erence.wolfram.com/mathematica/ref/Part.h=" target="_blank">http://refe=
rence.wolfram.com/mathematica/ref/Part.h=</a><br>
tml"><FONT face=Mathematica1>=EF=82=87</FONT>&l=
t;/A></P>=0A<P>&nbsp;</P>=0A<=<br>P>Im=
age::imgarray: The specified argument \[NoBreak]originalColorData\[NoBr=<=
br>eak] should be an array of rank 2 or 3 with machine-size numbers. <A =
href=<br>
="<a href="http://reference.wolfram.com/mathematica/ref/Image.html=
" target="_blank">http://reference.wolfram.com/mathematica/ref/Image.html=
</a>"><FONT face=M=<br>athematica1>=EF=82=87</FO=
NT></A></P>=0A<P>I will be very grateful if you co=
=<br>
uld help me.</P>=0A<P>Thanks a lot</P>=0A<P>Bes=
t regards</P>=0A<P>Maria</P>=<br>=0A<P>&n=
bsp;</P></B></FONT>=0A<DIV style="FONT-FAMILY=
: arial, helvetica, =<br>
sans-serif; FONT-SIZE: 13px"><FONT size=2 face=Tahoma>==
0A<HR SIZE=1>=<br>=0A<B><SPAN style="FONT-WEIGH=
T: bold">Da:</SPAN></B> J. Batista &lt;jbati=<br>=
<a href="mailto:sta800 at gmail.com">sta800 at gmail.com</a>&gt;<BR>&=
lt;B><SPAN style="FONT-WEIGHT: bold">A:</SPAN>&l=
t;/B> =<br>
<a href="mailto:mathgroup at smc.vnet.net">mathgroup at smc.vnet.net</a><BR&=
gt;<B><SPAN style="FONT-WEIGHT: bold">Inviato:</=
SP=<br>AN></B> Lun 2 agosto 2010, 13:02:00<BR><B><=
;SPAN style="FONT-WEIGHT: bold"=<br>
>Oggetto:</SPAN></B> [mg111445] Re: Problem with Min Max bet=
ween two functi=<br>ons<BR></FONT><BR>Maria/All, I just=
learned from a colleague that all equat=<br>ions on my message<BR>=
have double equal signs.&nbsp; Please note that is pr=<br>
obably due to a transmission<BR>error.&nbsp; All equations should=
only have=<br> a single equal sign.&nbsp; I'm<BR>retrans=
mitting my original message.<BR>R=<br>egards,<BR>J. Batista&l=
t;BR>On Sun, Aug 1, 2010 at 3:10 AM, J. Batista &lt;<A =<br>
href="mailto:<a href="mailto:jbatista800 at gmail.com">jbatista800@gm=
ail.com</a>" ymailto="mailto:<a href="mailto:jbatista800@gmai=
l.c">jbatista800 at gmail.c</a>=<br>om"><a href="mailto:jbatista80=
0 at gmail.com">jbatista800 at gmail.com</a></A>&gt; wrote:<BR>&l=
t;BR>&gt; Dean Maria, here is a =<br>
possible solution to your question.&nbsp; The white<BR>&gt; c=
ontour areas c=<br>an be<br> removed by approaching the output of Conto=
urPlot as an<BR>&gt; image, trea=<br>ting the task as one of im=
age processing (as suggested previously<BR>&gt; b=<br>
y Daniel Lichtblau).&nbsp; First, equate the original ContourPlot outpu=
t wi=<br>th a<BR>&gt; variable name, for example originalPlot =
== Out[1]&nbsp; (w=<br>here Out[1] is the<BR>&gt; Conto=
urPlot output cell).&nbsp; Alternatively, y=<br>
ou can simply select the<BR>&gt; ContourPlot, copy and then paste=
into the =<br>first line of the code sequence<BR>&gt; below in=
place of the variable orig=<br>inalPlot.&nbsp; I will now display th=
e four<BR>&gt; lines of the code seque=<br>
nce and then explain them afterwards.<BR>&gt;<BR>&gt; o=
riginalColorData ==<br>= ImageData[originalPlot];<BR>&gt;&l=
t;BR>&gt; targetPixels == Position[o=<br>riginalColorData, ori=
ginalColorData[[180, 180]]];<BR>&gt;<BR>&gt; newColorD=
=<br>
ata == ReplacePart[originalColorData, targetPixels -&gt; {1., 1.,&l=
t;BR>&g=<br>t; 1.}];<BR>&gt;<BR>&gt; Image[new=
ColorData]<BR>&gt;<BR>&gt;<BR>&gt; The fi=<=
br>rst code<br>
line accomplishes the task of collecting and reading the<BR>&g=
t; ContourPl=<br>ot into computer memory as image data, in this case a ve=
ctor of RGB<BR>&gt;=<br> color values.&nbsp; Note that I =
place a semicolon at the end of this and o=<br>
ther<BR>&gt; lines of code in order to suppress the visual output=
of the co=<br>de sequence's<BR>&gt; result.&nbsp; This=
is because the vector is lengthy a=<br>nd will clog the notebook<BR&g=
t;&gt; unnecessarily.<BR>&gt; The second code li=<br>
ne establishes the pattern by which the portions of the<BR>&gt; p=
lot that y=<br>ou wish to alter are identified as a subset of the entire&=
lt;BR>&gt; original =<br>data set.&nbsp; The pattern is establ=
ished by entering the pixel<BR>&gt; co=<br>
ordinate of a representative target pixel that you wish to alter, in this&l=
t;B=<br>R>&gt; case [[180, 180]] being one of the pixels in the wh=
ite contour areas=<br>.&nbsp; You<BR>&gt; can determine an =
appropriate pixel coordinate by right-=<br>
clicking in the<BR>&gt; original ContourPlot output, selecting Ge=
t<br> Indices, and then guiding your<BR>&gt; cursor to a desire=
d location within=<br> the plot.<BR>&gt; The third code line =
replaces the pixel locations flagged=<br>
by the previous<BR>&gt; pattern search with new pixel data, in=
this case n=<br>ew RGB color values that<BR>&gt; you select.&a=
mp;nbsp; I have used the example =<br>of {1., 1., 1.} to illustrate chang=
ing<BR>&gt; from the semi-white color of=<br>
the original plot to a true white that matches<BR>&gt; the plo=
t background=<br>.&nbsp; Be sure to use decimal points as above when =
expressing<BR>&gt; colo=<br>r values for your pixels, as someth=
ing like {1, 1, 1} will not<BR>&gt; be u=<br>
nderstood correctly for this purpose.&nbsp; If you want to change the&l=
t;BR>&g=<br>t; semi-white color of your original plot to black, us=
e {0., 0., 0.}.<BR>&g=<br>t; The fourth and final line re-estab=
lishes the newly altered set of pixel<=<br>
BR>&gt; data as an image object, and displays the altered image.<=
BR>&gt;<BR=<br>>&gt; Hope this helps.<BR>&gt; =
Best Regards,<BR>&gt; J.<br> Batista<BR>&gt;<BR>=
;&gt;&nbsp; On Mon, Jul 26, 2010 at 6:37 AM, maria giova=<br>
nna dainotti &lt;<BR>&gt; <A href="mailto:<a href=
="mailto:mariagiovannadainotti at yahoo.it">mariagiovannadainotti at yahoo.it</=
a>"=<br> ymailto="mailto:<a href="mailto:mariagiovannadai=
notti at yahoo.it">mariagiovannadainotti at yahoo.it</a>">mariagiovannada=
inotti@ya=<br>
<a href="http://hoo.it/"; target="_blank">hoo.it</a></A>&gt; w=
rote:<BR>&gt;<BR>&gt;&gt; Dear Mathgroup,<BR>=
&gt;&gt; I hav=<br>e the following function<BR>&gt;&=
;gt; R1==1.029<BR>&gt;&gt; R2==3.89=<br>
2<BR>&gt;&gt; R3==8<BR>&gt;&gt; e1==250=
<BR>&gt;&gt; e2==11.8<BR=<br>>&gt;&gt; e3=
==80.5<BR>&gt;&gt; i==pi/12<BR>&gt;&gt;=
spherenear[x_,y=<br>
_]:==((R3^2-x^2-y^2)^(1/2))<BR>&gt;&gt; spherefar[x_,y_]:=
==-((R3^2-=<br>x^2-y^2)^(1/2))<BR>&gt;&gt; emptynear[x_=
,y_]:==Min[Re[spherenear[x,y]],=<br>Re[(R2^2-x^2)^(1/2)+y*Tan[i]]=&=
lt;BR>]<BR>&gt;&gt; emptyfar[x_,y_]:==Max=<br>
[Re[spherefar[x,y]],Re[-(R2^2-x^2)^(1/2)]+y*Tan[i]]<BR>&gt;&g=
t; jetnear[x_,=<br>y_]:==Min[Re[spherenear[x,y]],Re[(R1^2-x^2)^(1/2)+=
y*Tan[i]]]<BR>&gt;&gt=<br>; jetfar[x_,y_]:==Max[Re[sphe=
refar[x,y]],Re[-(R1^2-x^2)^(1/2)]+y*Tan[i]=<br>
]<BR>&gt;&gt;<BR>&gt;&gt;<br> f[x_,y_]:===
((jetnear[x,y]-jetfar[x,y])*e1+e3*(spherenear[x,y]-emptynea=<br>r[=<=
BR>x,y]+emptyfar[x,y]-spherefar[x,y])+e2*(emptynear[x,y]-jetnear[x,y]==
<br>+jetfar[x=<BR>,y]-emptyfar[x,y]))*Boole[-R1==EF==82==
=A3x==<br>
=EF==82==A3R1]+((emptynear[x,y]-=<BR>emptyfar[x,y])*e2+e3=
*(sphe=<br>renear[x,y]-emptynear[x,y]+emptyfar[x,y]-spherefa=<BR>=
r[x,y]))*(Boole[-R2=<br>&lt;x&lt;-R1]+Boole[R2&gt;x&gt;R1=
])+(spherenear[x,y]-spherefar[x,y])=<BR>=<br>
*e3*(Boole[x&lt;-R2]+Boole[x&gt;R2])<BR>&gt;&gt;<B=
R>&gt;&gt;<BR>&gt;&gt; Co=<br>ntourPlot[f[x,y],{=
x,-8,8},{y,-8,8},Contours==EF==82==AE{0,100,2=<br>00,300,=&=
lt;BR>400,500,600,700,800,900}]<BR>&gt;&gt;<BR>&=
gt;&gt; From the pi=<br>
cture you can see there is a white contours that results a bi=<BR>t=
<BR>&g=<br>t;&gt; odd, I<BR>&gt;&gt; think th=
at it comes out from the introduction of=<br> the Min and Max.<BR&g=
t;&gt;&gt; I would like to remove this white contour. Co=<br>
uld you help me?<BR>&gt;&gt; Thanks a lot for your attention&=
lt;BR>&gt;&gt; Ch=<br>eers<BR>&gt;&gt;<br> Mar=
ia<BR>&gt;&gt;<BR>&gt;&gt;<BR>&gt;<=
;BR><BR><BR></DIV></DIV></div><br>=
=0A=<br>
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