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fractint-digest        Wednesday, May 13 1998        Volume 01 : Number 201




----------------------------------------------------------------------

Date: Tue, 12 May 1998 21:47:31 -0700
From: "Jay Hill" <[e-mail]>
Subject: Re: (fractint) useless tip #314

> From: Angela Wilczynski <[e-mail]>
>
> OK OK OK .....I can't stand it....what is periodicity??
>
> Like a good fractint user I checked out the documentation first and it says
>
> Note that fractint normally uses periodicity checking to speed up
>   bifurcation computation.  However, in some cases a better quality image
>   will be obtained if you turn off periodicity checking with
>   "periodicity=no"; for instance, if you use a high number of iterations
>   and a smooth colormap.
>

Hi wizzle,

Consider the normal simple Mandelbrot set. The solid components correspond
to regions where the orbit (the iteration samples) eventually settle into a
periodic pattern and therefore never can escape. Fractint uses periodicity
checking to stop earlier than maxiter. It usually works nicely.  However on
some other fractal formula you can find checkerboard like patterns in spots
where the periodicity checking stopped to early. This parameter allows you
to control this. Turning it off will cause a longer calculation time but give a

possibly more correct image.

> Now I'm sure this is English because I recognize many of the words. I also
read
> the longer explanation on p. 140 of the documentation (which I have as a .doc
> file thanks to a hint from this list). But I STILL don't know what <g> or any
of
> those other things do for me.  Could someone explain how it affects an image
> and when it should be used? and how?
>

I hope my English is understandable.

Jay

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Date: Tue, 12 May 1998 22:01:14 -0700
From: Mark Christenson <[e-mail]>
Subject: (fractint) par restore menu wish list item

I haven't posted any pars lately because a fit of conscience;
I would like to provide comment on other work posted here
before going back into egocentric mode, but I'm wading through
the April '98 par collection.  Hence the wish list item.  Since
I get the feeling that v20.0 is not far off I wanted to get my bid
in.  If the feature already exists, let me know where and accept
my apology in advance.

What I would like, to help me get a handle on massive par
collections, is the ability to delete pars from the [@/2] par
restore menu.  This would be a great labor saver for me, and,
I suspect, others.


Thanks, Bud


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Date: Wed, 13 May 1998 01:23:32 -0400 (EDT)
From: Jim Muth <[e-mail]>
Subject: Re: (fractint) par restore menu wish list item

At 10:01 PM 5/12/98 -0700, you wrote:
>
>What I would like, to help me get a handle on massive par
>collections, is the ability to delete pars from the [@/2] par
>restore menu.  This would be a great labor saver for me, and,
>I suspect, others.
>

Good idea.  It would be convenient to also be able to delete
unwanted fractal images from the menu.


Jim Muth
[e-mail]


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Date: Wed, 13 May 1998 00:44:49 -0500 (CDT)
From: [e-mail] (Paul and/or Joyce Carlson)
Subject: (fractint) One revised, one new par

comment {
I've posted this formula before along with a different
coloring for the first par.  The first par in this version
looks much better (IMHO).  The second par is a zoom into
the first.

I'm including the formula again.  I'm not sure this is
necessary.  I will be posting pars in the future that
use formulas I've already posted and I'm wondering if it
is worthwhile to post the formula each time.  I have a
feeling that if I don't post the formula each time, there
will be requests for it that may take up more bandwidth
than the formula.  What do you think?

Paul Carlson
}

frm:Mandel_3Way {; Copyright (c) Paul W. Carlson, 1998
    ; real(p1) = escape circle radius squared
    ; imag(p1) = number of iterations to skip
    ; real(p2) = number of color ranges
    ; imag(p2) = number of colors in a range
    z = w = k = iter = range_num = bailout = 0
    c = pixel
    prev_modw2 = 1.0e20
    rad2 = real(p1)
    skip_iters = imag(p1)
    num_ranges = real(p2)
    colors_in_range = imag(p2)
    colors_in_range_1 = colors_in_range - 1:
    w = w * w + c
    IF ((prev_modw2 < rad2) && (|w| > rad2))
    bailout = 1
    IF (iter > skip_iters)
        k = 1
    ENDIF
    index = colors_in_range_1 * (rad2 - prev_modw2) / rad2
    z = (index + range_num * colors_in_range + 1) * k
    ENDIF
    prev_modw2 = |w|
    range_num = range_num + 1
    IF (range_num == num_ranges)
    range_num = 0
    ENDIF
    iter = iter + 1
    z = z - iter
    bailout == 0 && |w| <= rad2
}

mnd3wm01           { ; Copyright (c) Paul W. Carlson, 1998
  reset=1960 type=formula formulafile=mand3way.frm
  formulaname=mandel_3way passes=t
  corners=-1.994336436926/-1.994329835296/4.401086666311e-006/-\
  4.401086667012e-006/-1.994329835296/-4.401086667012e-006
  params=4/21/2/125 float=y maxiter=2000 inside=253 outside=summ
  colors=000OOO<35>xxxzzzyyy<37>OOO000<46>000aG0<35>yx0zz0zy0<3\
  7>aG0000<51>000
  }

mnd3wm02           { ; Copyright (c) Paul W. Carlson, 1998
  reset=1960 type=formula formulafile=mand3way.frm
  formulaname=mandel_3way passes=t
  corners=-1.994334425434/-1.994331864002/1.707621626269e-006/-\
  1.70762162697e-006/-1.994331864002/-1.70762162697e-006
  params=4/21/2/125 float=y maxiter=2000 inside=253 outside=summ
  colors=000OOO<35>xxxzzzyyy<37>OOO000<46>000aG0<35>yx0zz0zy0<3\
  7>aG0000<51>000
  }


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------------------------------

Date: Wed, 13 May 1998 11:07:37 +0200
From: Joe Pearson <[e-mail]>
Subject: RE: (fractint) useless tip #314

Les, this solves the problem of typing peridocityi correctly, but as
Damien pointed could still be several keystrokes to get from the par you
were busy with to commands.par (and back again), and most importantly it
still doesn't give a way of finding out the existing setting of
periodicity.

Joe's idealistic design, no copyright reserved:
call up an "inside calculations options" screen.  periodicity is one of
the values displayed.  change if necessary.  if confused, hit F1 and go
directly to the explanation of what & why periodicity

or, when you've heard of periodicity but don't know where it's set, hit
F1, select a search function with the help system, type "periodicity"
(or just "period") and search, to get the same help information.

As I said before, this doesn't mean a windows implementation.  I'd like
to see this stuff in DOS asap (v20.1?).

Joe

>----------
>From:     Les St Clair[SMTP:[e-mail]]
>Damien M. Jones  wrote...
>
>>Because pressing a single key for it, or even two keys, is a lot easier
>>than typing p e r i o d i c i t y = 0  :)  I can't count the number of
>>times I've misspelled it and received a complaint from FractInt.
>
>and Joe Pearson wrote...
>
>>>as well as all the options that are accessible by 'g' only<<
>
>Any command line parameter can be written in par format, so you could write a
>parameter file with all your favourite command line options in it.
>To access the options just hit <2> and choose the one you want from the list.
>
>Here's a simple example...
>
>==== cut

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------------------------------

Date: Wed, 13 May 1998 11:22:12 +0200
From: Joe Pearson <[e-mail]>
Subject: RE: (fractint) par restore menu wish list item

>At 10:01 PM 5/12/98 -0700, you wrote:
>>What I would like, to help me get a handle on massive par
>>collections, is the ability to delete pars from the [@/2] par
>>restore menu.  This would be a great labor saver for me, and,
>>I suspect, others.
>Good idea.  It would be convenient to also be able to delete
>unwanted fractal images from the menu.
>Jim Muth
And Angela said: One area where I have a real problem with Fractint is
in the area of file
management.

I'd like to see *all* fractint's file menus giving you the ability to
delete, rename and make copies of existing files.  It would be off-topic
for me to mention the name of an operating system that shows one way of
doing this.

>Joe
>

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------------------------------

Date: Wed, 13 May 1998 11:33:31 +0200
From: Joe Pearson <[e-mail]>
Subject: RE: (fractint) par restore menu wish list item

>
>I'd like to see *all* fractint's file menus giving you the ability to
>delete, rename and make copies of existing files.  It would be off-topic
>for me to mention the name of an operating system that shows one way of
>doing this.
>
>>Joe
>>
That's an excellent idea Joe :-)
You should also have said the file menus should enable you to see
(and/or sort by) the size and date of the files.
As a fractint unique it would also be nice to see the calc time, and
whether a saved image has completed calculation ... but I can wait until
v20.2 for these features!
>
>

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------------------------------

Date: Wed, 13 May 1998 12:11:40 +0200
From: Joe Pearson <[e-mail]>
Subject: RE: (fractint) fractint v20 and beyond

>> Can we please terminate this thread?
>
>Well said, sir!
>
>I think the protagonists have said their pieces; let anyone else who
>has an opinion voice it, and then let's let this thread die.

Maybe it's just the consultant in me (refer to Dogbert for truth about
consultants) but I wouldn't like the thread to die just as a load of
opinions voiced.  Can we have a response from the Stone Soup group?
What of the useful features discussed in this thread can practically be
implemented in fractint?  How long, and how many other programmers need
to volunteer?

Thanks - Joe

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------------------------------

Date: Wed, 13 May 1998 13:12:07 +0200
From: Joe Pearson <[e-mail]>
Subject: (fractint) Double Exposure

Paul,

Just played with these pars.  Very impressive effect: more of a 3D
surface effect with colour stripes not following the contours, for a
change.  Perhaps I must pay more attention to posted formulae.

One plea: could you put the calculation time in your first comment line?
 Some of them are just too long for my little machine to calculate, but
I don't want to ignore the entire set just because of this.

thanks
Joe

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------------------------------

Date: Wed, 13 May 1998 06:49:55 -0700 (PDT)
From: "James R. McKenzie" <[e-mail]>
Subject: Re: (fractint) Some boundary trace pars

I don't have any of the formulae for these Pars.  Could someone please
send them to me ASAP?



                                           T H A N K   Y O U


                                          James R. McKenzie



                                [e-mail]

_________________________________________________________
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Date: Wed, 13 May 1998 08:00:36 -0700
From: "Angela Wilczynski" <[e-mail]>
Subject: Re: (fractint) par restore menu wish list item

I delete my gifs with a viewer in Windows.....actually makes more sense to me
since I can see the image prior to deleting it. However, I would like a way to
delete maps from Fractint, since I can only see them in Fractint.

Actually...what I would REALLY like is a little utility that allows me to see
the maps using a Windows graphics program that accepts Photoshop plug ins like
Paintshop Pro or Corel Draw. More than anything I am after the ability to
compare two or more maps before deciding which one to dump.

Angela

Jim Muth wrote:
>
> At 10:01 PM 5/12/98 -0700, you wrote:
> >
> >What I would like, to help me get a handle on massive par
> >collections, is the ability to delete pars from the [@/2] par
> >restore menu.  This would be a great labor saver for me, and,
> >I suspect, others.
> >
>
> Good idea.  It would be convenient to also be able to delete
> unwanted fractal images from the menu.
>

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Date: Wed, 13 May 1998 10:45:25 -0400
From: "RJ Corradino" <[e-mail]>
Subject: (fractint) DarkEye finished par

I finally have what I've been striving for.  These are the two tweaks I made
since yesterday.  I personally prefer 8, but I thought it would be fun for
you guys to see the progression from the first attempt(which was crappy at
best) into #8(which is half-way decent).  For me, it began looking good
after I zoomed in a bit more.  Then everything fell into place in DarkEye6,
when I added the gold 'veins.'  From there, #7 is that same map, cycled a
tad.  In 8 I only needed to darken a few segments.  Have fun with it.  I
certainly did.

DarkEye7           { ; ;
                     ; Image Copyright May 12 1998 by RJ Corradino
                     ; e-mail to:[e-mail]
  reset=1960 type=tim's_error function=sqr
  center-mag=-0.91061682690876760/+0.12383459253096580/206289.8/0.02192/-4\
  5.859/79.094 params=0/0 float=y maxiter=5000 bailout=5000
  colors=R38<52>S38S38S38T38T38T38<22>W37W37X37Y37Z37_37<6>Y36<2>m36<4>V35\
  <6>S34R34Q34P34O34<12>R84R94R94SA4SA4SB4<62>mf0mf0le0<59>000
  }

DarkEye8           { ; ;
                     ; Image Copyright May 12 1998 by RJ Corradino
                     ; e-mail to:[e-mail]
  reset=1960 type=tim's_error function=sqr
  center-mag=-0.91061682690876760/+0.12383459253096580/206289.8/0.02192/-4\
  5.859/79.094 params=0/0 float=y maxiter=5000 bailout=5000
  colors=R38<52>S38S38S38T38T38T38<22>W37W37X37Y37Z37_37<2>Z37P36K36J36E36\
  E36S36m36<4>V35<6>S34R34Q34P34O34<12>R84R94R94SA4SA4SB4<62>mf0mf0le0<59>\
  000
  }


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Date: Wed, 13 May 1998 07:57:19 PDT
From: "john Weeks" <[e-mail]>
Subject: Re: (fractint) Superior Images?

davides writes:
>In what way, if I may ask, are the images of which you write superior
to
>the images generated from fractint?
>TIA

There has been some confusion about my post.  I hardcode my images into
a compiler and have the advantage of 32-bit interface(sound, color,
etc.).  I have NOT found a fractal generator that works better than
Fractint.  The point of my post was that Fractint is the most useful and
powerful generator (despite its limitations), because of the number of
people who use it and contribute to it's libraries.
              sorry for any confusion,
                  john Weeks
  If you have any further questions about what I've been doing with
Fractals you can reach me at:
[e-mail]


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Date: Wed, 13 May 1998 08:32:39 -0700
From: "Angela Wilczynski" <[e-mail]>
Subject: Re: (fractint) Oh joy, another newbie

RJ Corradino wrote:
>
> Angela-
>
> Thanks very much for the advice and the warm welcome.  I'll be sure to try
> out all of those ideas.
>
> >By and large, it is the construction of the map rather than the individual
> >colors selected (or their sequence) that seems to determine if the map will
> be
> >generally useful or only in particular situations.
>
> What do you mean by construction?  Construction and sequence seem to be the
> same thing to me(both seem to refer to the overall layout of the colors)...
>
> Thanks,
> -RJ


I'm replying to the list because I think that is a darn good question. Graphics
are hard to learn because they are difficult to explain.

For me construction and sequence are NOT synonymous.  Sequence means red follows
blue then comes yellow, etc. and color range is the lightest to darkest values
of a particular sequence of colors.

Construction has to do with the exact placement of the color range.  Paul
Carlson's maps are very carefully constructed to fit his formulas, so they are
an excellent illustration of what I mean.  Below are a par and two maps,
flx-blk2.map is a variation of Paul's original map.  Paul's map is constructed
as follows:

1.  Each color range is 30 colors long.
2.  The color range starts with the lightest value and ends with the darkest
value.
3.  There is no transitioning gradient to the next color range - the change is
abrupt from one color to the next
4.  The lightest values used are bright, saturated hues (as opposed to midtones,
pastels, greyed-tones, etc.)
5.  Black is used as a "filler" at the end of the colors, except that the color
in position #253 is red (this is because paul's formula calls for the inside
color to be #253 as you can see in my par.)


wiz_spiralvalley   { ; AVW 5/9/98 new color map flx-blk2.map
                     ; from a Paul Carlson formula
  reset=1960 type=formula formulafile=may98.frm
  formulaname=spir_stalks_jul
  corners=-0.53691921/-0.479572948/0.802008049/0.780418614/-0.505911627/0.\
  760664605 params=-0.36665/0.11573/0.02/0/8/30 float=y maxiter=251
  inside=253 outside=summ
  colors=000zzc<28>cKA000<28>0000wz<27>03C00A000<28>000fOz<28>K0F000<28>00\
  0KKU<26>333222111000000<42>000
  }

map_construction   { ; original map by Paul Carlson
  colors=000fOz<28>I0Kz0f<28>O08z88<28>O00zW0<28>c40zz0<28>aG00zR<28>0C40z\
  z<28>0CCGGz<28>00O000<10>000z88000000
  }

Knowing how Paul constructs his maps and what this construction does, it was
easy for me to vary his map to get the result I wanted........a series of colors
for a VERY busy image.  If you've seen my color hints, you might remember that I
said to use a map with black on a busy image to bring out it's structure. So,
what I did was to replace every other 30 color range with black. Then I also
changed the remaining colors to the "wizzle" pallet. But note that my map also
makes the color ranges go from lightest to darkest.

If you check out other maps Paul uses or my variations, you will see that his
construction is exactly the same Except, the color range starts with the darkest
color, goes to the lightest color 10 colors "in"....then goes back to the
darkest color again. Different "construction." As a matter of fact, Paul uses
the same colors in both types of maps, but the placement of the colors is vastly
different and very significant.

Other methods of "construction" I've seen are:

1.  Start with black......place the lightest value to end 32 colors away and
make a gradient...repeat
2.  Start with white.....and do the same thing
3.  Start with the darkest value of color #1.....put the same color at square #
126.....put the lightest color midway between and make a gradient. Repeat for
color #2 - this is a standard two-color map (I have samples in the collection)

Hope this helps

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Date: Wed, 13 May 1998 13:32:43 -0400
From: "RJ Corradino" <[e-mail]>
Subject: RE: (fractint) par restore menu wish list item

Mark wrote;
> What I would like, to help me get a handle on massive par
> collections, is the ability to delete pars from the [@/2] par
> restore menu.  This would be a great labor saver for me, and,
> I suspect, others.

I too would find this useful.  What would be even better for me is a windows
par/frm manager.  Perhaps a 3 pane window, the first frame listing all the
pars and frms in a givin directory, the second having a list of all the
entries in the par, and the third showing the actual text of the selected
entry.  This way you could drag and drop fractals from par to par(I've been
copying the list pars into a generic today.par , run partobat,  and then
cutting and pasting the ones I like into permanent pars), move a formula
thats embedded in a par to a frm file, delete whatever you no longer need,
add comments, or whatever else.

Some other nice features, which I imagine would be harder to add:
1- Ability to right click highlighted text use the windows 'send to' feature
to add the text to a par.

2- Double click a par to run it in fractint.

Are there any programs like this?  It'd be a great organization tool!


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Date: Wed, 13 May 1998 14:51:17 -0400
From: [e-mail]
Subject: (fractint) gallet-6-08 formula

Here are some fun things you can do with Silvie Gallet's formula
gallet-6-08.

Bill Decker (http://www.geocities.com/SoHo/Studios/1450/)

frm:Gallet-6-08 { ; Sylvie Gallet, 1996
z = pixel , x = real(z) , y = imag(z) :
x1 = x - p1*fn1(y + fn2(p2*x) / fn3(p3*y))
y1 = y - p1*fn1(x + fn2(p2*y) / fn3(p3*x))
x = x1 , y = y1
z = x + flip(y)
|z| <= 1024
}

frame2             { ;  (c) 1998 Bill Decker
                     ; Abstract form
  reset=1960 type=formula formulafile=frac_ml.frm
  formulaname=Gallet-6-08 function=cos/ceil/atanh
  center-mag=-0.00176519/0.00831852/0.6206756/1/45 params=2/1/5/0/7/6
  float=y maxiter=255 potential=256/200/0 decomp=256 periodicity=0
  colors=MC6000NC6<8>W67X57_96<9>zm0<6>wa0w_0vY0uW0<14>NC9KAAKBA<19>1X10Z0\
  0Z1<29>68a77c77c<48>44K44K64J<12>`96cA5dC5<14>zm7<14>XD6UA5TA5<13>752541\
  551<14>BU3<12>EH4FG4FF4GE5HF6<4>LD6 cyclerange=0/255
  }

lamp               { ;  (c) 1998 Bill Decker
                     ; Just a small piece of something larger
  reset=1960 type=formula formulafile=frac_ml.frm
  formulaname=Gallet-6-08 function=cos/ceil/atanh
  center-mag=+0.00897415973749732/+0.00899740861882518/1713.816/1/44.999
  params=10/10/10/10/10/10 float=y maxiter=255 potential=256/200/0
  decomp=256 periodicity=0
  colors=OB6<7>W67X57_96<9>zm0<6>wa0w_0vY0uW0<14>NC9KAAKBA<19>1X10Z00Z10Y2\
  0X4<3>4U55T56R5<9>GE6HC7IC7<12>eE3gF2iE2<10>x81z70z70<42>P7B<4>`96cA5dC5\
  <14>zm7<14>XD6UA5TA5<13>752541551<14>BU3<12>EH4FG4FF4GE5HF6<2>JE6KE6LD6M\
  C6000NC6 cyclerange=0/255
  }

cut-lace           { ;  (c) 1998 Bill Decker
                     ; Abstract form
  reset=1960 type=formula formulafile=frac_ml.frm
  formulaname=gallet-6-08 function=cosh/sinh/sinh
  center-mag=0.16845/0.0436802/0.3065529/1/44.998 params=5/1/0/5/1/10
  float=y maxiter=255 potential=255/20/0 decomp=256 periodicity=0
  colors=@satin.map cyclerange=0/255
  }

textures3          { ;  (c) 1998 Bill Decker
                     ; Abstract form
  reset=1960 type=formula formulafile=frac_ml.frm
  formulaname=gallet-6-08 function=log/recip/exp
  center-mag=-0.949972/-0.86828/0.1671804/1/44.997 params=5/5/0/5/1/10
  float=y maxiter=255 potential=255/100/0 decomp=256 periodicity=0
  colors=rrZvvazzc<4>ffSZZN<8>000<15>czc<4>SfSN_N<2>HRHFOFDKDAGA<3>000<15>\
  zmU<5>bVJZSHWPFTME<2>IF9FC8C96864432000<15>zZF<14>000000<15>cUe<15>000<1\
  5>zcm<15>000<15>mmz<15>000<15>mzz<6>T__JOO<3>688344233122000<12>nnX
  cyclerange=0/255
  }

pictureframe       { ;  (c) 1998 Bill Decker
                     ; Viewed through glass bricks
  reset=1960 type=formula formulafile=frac_ml.frm
  formulaname=gallet-6-08 function=sqr/cos/tanh
  center-mag=0.00421051/-0.00680109/3.286374/0.9999/90
  params=2/2/2/2/2/30 float=y maxiter=255 potential=255/200/0
  decomp=256 periodicity=0
  colors=6C1<9>BS3BU3BT3<11>EH4FG4FF4GE5HF6<2>JE6KE6LD6MC6000NC6<8>W67X57_\
  96<9>zm0<6>wa0w_0vY0uW0<14>NC9KAAKBA<19>1X10Z00Z1<29>68a77c77c<48>44K44K\
  64J<12>`96cA5dC5<14>zm7<14>XD6UA5TA5<13>752541551<2>6A1
  cyclerange=0/255
  }

butterflys         { ;  (c) 1998 Bill Decker
                     ; Part of a larger pattern
  reset=1960 type=formula formulafile=frac_ml.frm
  formulaname=gallet-6-08 function=cosxx/atan/asin
  center-mag=9.74691/9.97431/0.1733668/1/44.996 params=2/2/2/2/2/2
  float=y maxiter=255 potential=255/200/0 decomp=256 periodicity=0
  colors=6C1<10>BU3<12>EH4FG4FF4GE5HF6<2>JE6KE6LD6MC6000NC6<8>W67X57_96<9>\
  zm0<6>wa0w_0vY0uW0<14>NC9KAAKBA<19>1X10Z00Z1<27>6A_69`68a77c77c77c<47>44\
  K44K64J<12>`96cA5dC5<14>zm7<14>XD6UA5TA5<11>A629527525415515716816A1
  cyclerange=0/255
  }




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------------------------------

Date: Wed, 13 May 1998 12:07:11 -0700 (MST)
From: Kerry Mitchell <[e-mail]>
Subject: (fractint) Conic Sections coloring methods (long)

Warning!  This coloring method is not terribly user-friendly, because it
is powerful.  There are 6 parameters to specify the coloring, leaving none
for the Julia parameters, which are hard-coded.  But, a "help" formula is
included to guide your explorations.  Have fun!

- -------------------------------------------------------------------------------
Kerry Mitchell
[e-mail]
- -------------------------------------------------------------------------------

comment { ; narrative copyright Kerry Mitchell 12may98

Conic Sections

Conic sections are sections of cones.  Specifically, take a double-ended
cone, like 2 funnels placed tip-to-tip.  Then, form the intersection of
the (double) cone with a plane.  The intersection, usually one or two
curves, is a conic section.  Conic sections can be a point, one line, two
lines, a parabola, an ellipse, a circle, or a hyperbola, depending on the
orientation of the plane relative to the cone.  Analytically, they can all
be expressed by the same formula:

Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0,

where the parameters A through F determine the shape of the section, and x
and y are the 2 spatial coordinates.  For example, the line y=x can be
represented as

x - y = 0, or
A = 0, B = 1, C = 0, D = -1, E = 0, F = 0.

A circle centered at (1,0) with a radius of 2 would have the equation

(x - 1)^2 + y^2 = 4, or
x^2 - 2x + y^2 - 3 = 0, giving
A = 1, B = -2, C = 1, D = 0, E = 0, F = -3.

How can these shapes be used to render fractals?  First off, the "conic"
formula simply draws the section determined by the six parameters A - F,
to give the user an idea of how the parameter choices affect the section
generated.

The standard escape-time rendering method stops iterating when the
magnitude of the iterate is larger than some threshold.  Since the iterate
is a complex number, this means that the escape condition can be viewed in
terms of a 2-dimensional shape.  Specifically, if x and y represent the
real and imaginary parts of the iterate and the threshold level is r, then
escape occurs when:

x^2 + y^2 > r^2

which just means that the orbit leaves a circle of radius r, centered at
the origin.  Clearly, the circle could be replaced by any closed curve, so
long as it is large enough to enclose the fractal.  Open curves can also
be used as a bailout condition, so long as the appropriate side of the
curve is chosen.

This opportunity, changing the shape of the escape boundary, is coded into
the "conic-bail" formulas.  The six parameters (real and imaginary parts
of p1, p2, and p3) are used to determine a section; iteration ceases when
the orbit leaves this section.  (In actuality, the escape condition is
that the quantity

Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F

become larger than 0.  Q=0 is the equation for the escape boundary.)

(A note about the formulas:  Since 6 parameters are required to fully
specify a conic section, this leaves no room in the Fractint scheme to
interactively specify a Julia parameter.  Three versions of each Julia
formula are presented, to give a quick idea of the effect of Julia
parameter choice.  The user is free to change the specified parameter in
the formula, to fit their own needs.)

Fractint's "inside=zmag" coloring scheme is also built around the circle,
and thus, easily altered to a general conic section.  In the "conic-mag"
formulas, the standard bailout test is employed.  However, what is plotted
is the value of Q, as specified above.  Q=0 is the equation for the
boundary of the section, so this method effectively shows if the iterate
was inside or outside of the section when iteration ended.

The other two formula types record how the orbit interacts with the given
section.  In "conic-near", the distance to the section is tallied, and the
final color reflects the closest approach of the orbit to the section.  In
"conic-count", a counter is incremented for each iteration where the orbit
is inside the section.  This counter is scaled by the total number of
iterations for that pixel, and this fraction is converted into an angle
for "decomp=" plotting.

The standard forms of the conic sections are given below.  With a bit of
algebraic twiddling, they can be transformed into the general form, for
use in the Fractint formulas.

point (h,k)  x=h, y=k; or (x-h)^2 + (y-k)^2 = 0 [circle of radius 0
centered at (h,k)].

vertical line through (h,k):  x=h

non-vertical line with slope m, through (h,k):  y-k = m*(x-h)

parabola, opening up or down, with vertex at (h,k):
y-k = 4*p*(x-h)^2 [p determines width; +/up, -/down]

parabola, opening left or right, with vertex at (h,k):
x-h = 4*p*(y-k)^2 [p determines width; +/right, -/left]

ellipse centered at (h,k), semimajor axis alpha, semiminor axis beta:
(x-h)^2/alpha^2 + (y-k)^2/beta^2 = 1

circle centered at (h,k), with radius r:
(x-h)^2 + (y-k)^2 = r^2

hyperbola centered at (h,k), semimajor axis alpha, semiminor axis beta,
opening left/right:  (x-h)^2/alpha^2 - (y-k)^2/beta^2 = 1

hyperbola centered at (h,k), semimajor axis alpha, semiminor axis beta,
opening up/down:  (y-k)^2/beta^2 - (x-h)^2/alpha^2 = 1

coordinate rotation, from (u,v) to (x,y), through an angle theta:
u = x*cos(theta) + y*sin(theta)
v = -x*sin(theta) + y*cos(theta)
[rotating sections is how to generate non-zero E parameters]
}

tilted-ellipse     { ; copyright Kerry Mitchell 12may98
                     ; test case for conic formula
  reset=1960 type=formula formulafile=fractint.frm
  formulaname=conic center-mag=0/0/0.6666667 params=\
  1.5/-3/0.75/-1.25/1.25/1 float=y maxiter=256 inside=0
  decomp=256 periodicity=0 colors=000<61>y99zAAzAAzBB\
  <61>zzzzzzzyz<60>i1ih0hh0hg0g<61>000 cyclerange=0/255
  }

julia-circle       { ; copyright Kerry Mitchell 12may98
                     ; test case for conic-near formula
  ; colors by nearest orbit approach to conic section
  reset=1960 type=formula formulafile=fractint.frm
  formulaname=conic-near_juli center-mag=0/0/0.6666667
  params=1/0/1/-2/0/0.5 float=y maxiter=256 inside=0
  decomp=256 periodicity=0 colors=PAb<21>ZKj_Kk_Lk`Mk\
  <13>gUpgVphWphXpiXq<33>zyzzzzzyz<31>jZqiYqiXqhXphWp\
  gVp<25>WGhVFgVFgUEfUEf<18>K6_K6ZJ6ZJ6YI5Y<3>G4WG4WF\
  4VF3U<14>71M71L61K61J50I<3>30F30E20C20B<2>006000006\
  108<4>30F40G40H50I50I61J<8>A2QB2QB2RC2RC2S<5>F4VG4W\
  G4WH4W<14>O9b cyclerange=0/255
  }

julia-hyperbola    { ; copyright Kerry Mitchell 12may98
                     ; test case for conic-mag formula
  ; std bailout--colors by "amount" of conic section
  reset=1960 type=formula formulafile=fractint.frm
  formulaname=conic-mag_jul2 center-mag=0/0/0.4644195
  params=0.2/0/-0.2/0/0/0 float=y maxiter=256 inside=0
  decomp=256 periodicity=0 colors=000<40>x00z00z00<40>\
  zy0zz0zz1<39>zzxzzzzzz<40>1zz0zz0yz<39>02z00z00z<41>\
  000 cyclerange=0/255
  }

julia-ellipse      { ; copyright Kerry Mitchell 12may98
                     ; test case for conic-count formula
  ; colors by how often orbit is outside of conic section
  reset=1960 type=formula formulafile=fractint.frm
  formulaname=conic-count_jul16 center-mag=0/0/1/1/90
  params=1/-2/2/-1/-2/-2 float=y maxiter=256 inside=0
  decomp=256 periodicity=0 colors=Jzz<8>1zz0yz0wx<28>J2G\
  K0FK0F<30>b0Tc0Ud2V<29>yyyzzzxzz<29>1zz0yz0wx<28>J2GK0\
  FK0F<30>b0Tc0Ud2V<29>yyyzzzxzz<19>Lzz cyclerange=0/255
  }

mandel-parabola    { ; copyright Kerry Mitchell 12may98
                     ; test case for conic-bail formula
  ; bails out when orbit goes outside of conic section
  reset=1960 type=formula formulafile=fractint.frm
  formulaname=conic-bail_man center-mag=-1.77035703032\
  444400/+0.00538871966354182/3247556 params=1/3.5/0/1\
  /0/-16 float=y maxiter=1000 inside=0 decomp=256
  periodicity=0 colors=y00<40>000<43>xx0zz0yy0<39>2200\
  00010<40>0z0<40>020000100<41>z00 cyclerange=0/255
  }

frm:conic { ; Kerry Mitchell 12may98
;
;       draws conic sections, not fractals
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3), iter=-1:
        x=real(pixel), y=imag(pixel)
        t=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        t=log(cabs(t))
        z=cos(t)+flip(sin(t))
        iter>0
        }

frm:conic-mag_jul16 { ; Kerry Mitchell 12may98
;
;       colors Julia set by magnitude of "conic quantity"
;       Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to point near 1/16 disk
;       bailout set to 4
;
        cc=(0.2882,0.0106), zc=pixel, maxr=4, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc
        if ((|zc|>maxr)||(iter==maxit))
          iter=-1
          x=real(zc), y=imag(zc)
          t=x*(a*x+b)+y*(c*y+d)+e*x*y+f
          z=cos(t)+flip(sin(t))
          end if
        iter>0
        }

frm:conic-bail_jul16 { ; Kerry Mitchell 12may98
;
;       Julia set, escapes "conic quantity" Q > 0
;       Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).
;       Julia parameter set to point near 1/16 disk
;
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        cc=(0.2882,0.0106), zc=pixel, iter=1, e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        if ((conic>0)||(iter==maxit))
          iter=-1
          z=zc
          end if
        iter>0
        }

frm:conic-count_jul16 { ; Kerry Mitchell 12may98
;
;       colors Julia set by how often orbit is outside section
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to near 1/16 disk
;       bailout set to 10^12
;
        cc=(0.2882,0.0106), zc=pixel, maxr=1e12, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3), kount=0, scale=6.2832:
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        if (conic>0)
          kount=kount+1
          end if
        if ((|zc|>maxr)||(iter==maxit))
          t=scale*kount/iter
          z=cos(t)+flip(sin(t))
          iter=-1
          end if
        iter>0
        }

frm:conic-near_jul16 { ; Kerry Mitchell 12may98
;
;       colors Julia set by nearest approach to conic section
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to near 1/16 disk
;       bailout set to 10^12
;
       cc=(0.2882,0.0106), zc=pixel, maxr=1e12, minr=maxr, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=|x*(a*x+b)+y*(c*y+d)+e*x*y+f|
        if (conic<minr)
          minr=conic
          end if
        if ((|zc|>maxr)||(iter==maxit))
          iter=-1
          t=log(minr)
          z=cos(t)+flip(sin(t))
          end if
        iter>0
        }

frm:conic-mag_jul2 { ; Kerry Mitchell 12may98
;
;       colors Julia set by magnitude of "conic quantity"
;       Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to -2
;       bailout set to 4
;
        cc=-2, zc=pixel, maxr=4, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc
        if ((|zc|>maxr)||(iter==maxit))
          iter=-1
          x=real(zc), y=imag(zc)
          t=x*(a*x+b)+y*(c*y+d)+e*x*y+f
          z=cos(t)+flip(sin(t))
          end if
        iter>0
        }

frm:conic-bail_jul2 { ; Kerry Mitchell 12may98
;
;       Julia set, escapes "conic quantity" Q > 0
;       Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).
;       Julia parameter set to -2
;
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        cc=-2, zc=pixel, iter=1, e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        if ((conic>0)||(iter==maxit))
          iter=-1
          z=zc
          end if
        iter>0
        }

frm:conic-count_jul2 { ; Kerry Mitchell 12may98
;
;       colors Julia set by how often orbit is outside section
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to -2
;       bailout set to 10^12
;
        cc=-2, zc=pixel, maxr=1e12, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3), kount=0, scale=6.2832:
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        if (conic>0)
          kount=kount+1
          end if
        if ((|zc|>maxr)||(iter==maxit))
          t=scale*kount/iter
          z=cos(t)+flip(sin(t))
          iter=-1
          end if
        iter>0
        }

frm:conic-near_jul2 { ; Kerry Mitchell 12may98
;
;       colors Julia set by nearest approach to conic section
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to -2
;       bailout set to 10^12
;
        cc=-2, zc=pixel, maxr=1e12, minr=maxr, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=|x*(a*x+b)+y*(c*y+d)+e*x*y+f|
        if (conic<minr)
          minr=conic
          end if
        if ((|zc|>maxr)||(iter==maxit))
          iter=-1
          t=log(minr)
          z=cos(t)+flip(sin(t))
          end if
        iter>0
        }

frm:conic-mag_juli { ; Kerry Mitchell 12may98
;
;       colors Julia set by magnitude of "conic quantity"
;       Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to (0,1)
;       bailout set to 4
;
        cc=(0,1), zc=pixel, maxr=4, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc
        if ((|zc|>maxr)||(iter==maxit))
          iter=-1
          x=real(zc), y=imag(zc)
          t=x*(a*x+b)+y*(c*y+d)+e*x*y+f
          z=cos(t)+flip(sin(t))
          end if
        iter>0
        }

frm:conic-bail_juli { ; Kerry Mitchell 12may98
;
;       Julia set, escapes "conic quantity" Q > 0
;       Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).
;       Julia parameter set to (0,1)
;
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        cc=(0,1), zc=pixel, iter=1, e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        if ((conic>0)||(iter==maxit))
          iter=-1
          z=zc
          end if
        iter>0
        }

frm:conic-count_juli { ; Kerry Mitchell 12may98
;
;       colors Julia set by how often orbit is outside section
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to (0,1)
;       bailout set to 10^12
;
        cc=(0,1), zc=pixel, maxr=1e12, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3), kount=0, scale=6.2832:
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        if (conic>0)
          kount=kount+1
          end if
        if ((|zc|>maxr)||(iter==maxit))
          t=scale*kount/iter
          z=cos(t)+flip(sin(t))
          iter=-1
          end if
        iter>0
        }

frm:conic-near_juli { ; Kerry Mitchell 12may98
;
;       colors Julia set by nearest approach to conic section
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       Julia parameter set to (0,1)
;       bailout set to 10^12
;
        cc=(0,1), zc=pixel, maxr=1e12, minr=maxr, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=|x*(a*x+b)+y*(c*y+d)+e*x*y+f|
        if (conic<minr)
          minr=conic
          end if
        if ((|zc|>maxr)||(iter==maxit))
          iter=-1
          t=log(minr)
          z=cos(t)+flip(sin(t))
          end if
        iter>0
        }

frm:conic-mag_man { ; Kerry Mitchell 12may98
;
;       colors Mandelbrot set by magnitude of "conic quantity"
;       Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       bailout set to 4
;
        zc=0, cc=pixel, maxr=4, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc
        if ((|zc|>maxr)||(iter==maxit))
          iter=-1
          x=real(zc), y=imag(zc)
          t=x*(a*x+b)+y*(c*y+d)+e*x*y+f
          z=cos(t)+flip(sin(t))
          end if
        iter>0
        }

frm:conic-bail_man { ; Kerry Mitchell 12may98
;
;       Mandelbrot set, escapes "conic quantity" Q > 0
;       Q = Ax^2 + Bx + Cy^2 + Dy + Exy + F
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).
;
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        zc=0, cc=pixel, iter=1, e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        if ((conic>0)||(iter==maxit))
          iter=-1
          z=zc
          end if
        iter>0
        }

frm:conic-count_man { ; Kerry Mitchell 12may98
;
;       colors Mandelbrot set by how often orbit is outside section
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       bailout set to 10^12
;
        zc=0, cc=pixel, maxr=1e12, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3), kount=0, scale=6.2832:
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=x*(a*x+b)+y*(c*y+d)+e*x*y+f
        if (conic>0)
          kount=kount+1
          end if
        if ((|zc|>maxr)||(iter==maxit))
          t=scale*kount/iter
          z=cos(t)+flip(sin(t))
          iter=-1
          end if
        iter>0
        }

frm:conic-near_man { ; Kerry Mitchell 12may98
;
;       colors Mandelbrot set by nearest approach to conic section
;       Ax^2 + Bx + Cy^2 + Dy + Exy + F = 0
;       A=real(p1), B=imag(p1), C=real(p2), D=imag(p2)
;       E=real(p3), E=imag(p3).  use decomp=256 coloring
;       bailout set to 10^12
;
        zc=0, cc=pixel, maxr=1e12, minr=maxr, iter=1
        a=real(p1), b=imag(p1), c=real(p2), d=imag(p2)
        e=real(p3), f=imag(p3):
        iter=iter+1, zc=sqr(zc)+cc, x=real(zc), y=imag(zc)
        conic=|x*(a*x+b)+y*(c*y+d)+e*x*y+f|
        if (conic<minr)
          minr=conic
          end if
        if ((|zc|>maxr)||(iter==maxit))
          iter=-1
          t=log(minr)
          z=cos(t)+flip(sin(t))
          end if
        iter>0
        }





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