[Jprogramming] Plotting complex lists

km km at math.uh.edu
Tue Sep 24 02:00:07 UTC 2013 

It is correct and better!

--Kip Murray

Sent from my iPad

> On Sep 23, 2013, at 6:00 PM, Henry Rich <HenryHRich at nc.rr.com> wrote:
> 
> ratio =: [: %~/ [: (>./ - <./) [: +. ,
> 
> untested
> 
> Henry Rich
> 
>> On 9/23/2013 6:16 PM, km wrote:
>> Verb  ratio  calculates  (max - min of imaginary parts) % (max - min of real parts)  , thus:
>> 
>>     ratio =: [: %~/ [: (>./ - <./)"1 [: |: [: +. ,
>> 
>> Sample use:
>> 
>>     data =: _1 0j_1 1 0j1 _1 ,: 0j_3 0j_1.5 0 0j1.5 0j3
>>  NB. square and straight line
>>     ratio data
>>  3
>>     'aspect 3' plot data  NB. Shows the square as a square
>> 
>> --Kip Murray
>> 
>> Sent from my iPad
>> 
>>> On Sep 21, 2013, at 10:14 AM, km <km at math.uh.edu> wrote:
>>> 
>>> I hope the following simpler example will clarify.  Notes explain what the ratio is.
>>> 
>>>     data1 =: _1 0j_1 1 0j1 _1 ,: _2 _1 0 1 2
>>>     NB. a square and a straight line
>>> 
>>>     data2 =: 0 { data1  NB. a square
>>> 
>>>     plot data1  NB. plot shows square as parallelogram
>>> 
>>>     ratio data1  NB. this ratio is (1 - _1) % 2 - _2, see plot
>>>  0.5
>>> 
>>>     'aspect 0.5' plot data1  NB. plot shows square as square
>>> 
>>>     plot data2  NB. plot shows square as a parallelogram
>>> 
>>>     ratio data2  NB. this ratio is (1 - _1) % 1 - _1, see plot
>>>  1
>>> 
>>>     'aspect 1' plot data2  NB. plot show square as square
>>> 
>>>     IFIPAD
>>>  1
>>>     VERSION
>>>  1.3 5
>>> 
>>> --Kip Murray
>>> 
>>> Sent from my iPad
>>> 
>>>> On Sep 21, 2013, at 8:26 AM, Raul Miller <rauldmiller at gmail.com> wrote:
>>>> 
>>>> Do you mean something like an average of the sum along the last dimension?
>>>> 
>>>> If so, how important is it that any excess precision gets discarded?
>>>> 
>>>> Thanks,
>>>> 
>>>> --
>>>> Raul
>>>> 
>>>>> On Sat, Sep 21, 2013 at 8:38 AM, km <km at math.uh.edu> wrote:
>>>>> Challenge: devise a verb  ratio  so that if
>>>>> 
>>>>>    data1 =: ((,-)2j1) ,: _1^(%~i:)60
>>>>>    data2 =: _1^(%~i:)60
>>>>> 
>>>>> then
>>>>> 
>>>>>    ratio data1
>>>>> 0.5
>>>>>    ratio data2
>>>>> 1
>>>>> 
>>>>> i.e., if  data  is complex plot data and  r  is the result of  ratio data  then circles in
>>>>> 
>>>>>    'aspect r' plot data
>>>>> 
>>>>> appear circular.
>>>>> 
>>>>> --Kip Murray
>>>>> 
>>>>> Sent from my iPad
>>>>> 
>>>>>> On Sep 21, 2013, at 4:08 AM, km <km at math.uh.edu> wrote:
>>>>>> 
>>>>>> Try
>>>>>> 
>>>>>>   'aspect 0.5' plot ((,-)2j1) ,: _1^(%~i:)60
>>>>>> 
>>>>>> and
>>>>>> 
>>>>>>   'aspect 1' plot _1^(%~i:)60
>>>>>> 
>>>>>> --Kip Murray
>>>>>> 
>>>>>> Sent from my iPad
>>>>>> 
>>>>>>> On Sep 21, 2013, at 3:15 AM, Bo Jacoby <bojacoby at yahoo.dk> wrote:
>>>>>>> 
>>>>>>>   'aspect 1' plot ((,-)2j1) ,: _1^(%~i:)60 NB. This makes the frame of the plot square, but it has different units on the two axes, so the circle is deformed. How do I make the circle circular?
>>>>>>>   plot ((,-)2j1),:_1^(%~i:)60 NB. This is different, but not correct either.
>>>>>>> 
>>>>>>> 
>>>>>>>> ________________________________
>>>>>>>> Fra: Linda Alvord <lindaalvord at verizon.net>
>>>>>>>> Til: programming at jsoftware.com
>>>>>>>> Sendt: 3:39 lørdag den 21. september 2013
>>>>>>>> Emne: Re: [Jprogramming] Plotting complex lists
>>>>>>>> 
>>>>>>>> 
>>>>>>>> Square isn't so square either.  Linda  (Yet again, life is a series of
>>>>>>>> approximations.)
>>>>>>>> 
>>>>>>>> -----Original Message-----
>>>>>>>> From: programming-bounces at forums.jsoftware.com
>>>>>>>> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of Bo Jacoby
>>>>>>>> Sent: Wednesday, September 18, 2013 9:52 PM
>>>>>>>> To: programming at jsoftware.com
>>>>>>>> Subject: Re: [Jprogramming] Plotting complex lists
>>>>>>>> 
>>>>>>>> Thanks Linda! The 'aspect 1' makes the plot square. Do you know how to make
>>>>>>>> circles circular?
>>>>>>>> 
>>>>>>>> 
>>>>>>>>> ________________________________
>>>>>>>>> Fra: Linda Alvord <lindaalvord at verizon.net>
>>>>>>>>> Til: programming at jsoftware.com
>>>>>>>>> Sendt: 2:48 torsdag den 19. september 2013
>>>>>>>>> Emne: Re: [Jprogramming] Plotting complex lists
>>>>>>>>> 
>>>>>>>>> 
>>>>>>>>> Try replacing the last line with:
>>>>>>>>> 
>>>>>>>>> 'aspect 1' plot circle,ellipse,:hyperbola
>>>>>>>>> 
>>>>>>>>> Linda
>>>>>>>>> 
>>>>>>>>> -----Original Message-----
>>>>>>>>> From: programming-bounces at forums.jsoftware.com
>>>>>>>>> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of Bo
>>>>>>>>> Jacoby
>>>>>>>>> Sent: Tuesday, September 17, 2013 12:00 PM
>>>>>>>>> To: programming at jsoftware.com
>>>>>>>>> Subject: Re: [Jprogramming] Plotting complex lists
>>>>>>>>> 
>>>>>>>>> One benefit of using complex numbers is that you may forget about
>>>>>>>>> trigonometry.
>>>>>>>>> 
>>>>>>>>>   load'plot'
>>>>>>>>>   circle=._1^n=.(%~i:)60
>>>>>>>>>   ellipse=.(circle*-.a)+(+circle)*a=.0.8
>>>>>>>>>   hyperbola=.-:((+%)j.(-%))^n
>>>>>>>>>   plot circle,ellipse,:hyperbola
>>>>>>>>> 
>>>>>>>>> 
>>>>>>>>> 
>>>>>>>>> 
>>>>>>>>> 
>>>>>>>>> 
>>>>>>>>>> ________________________________
>>>>>>>>>> Fra: km <km at math.uh.edu>
>>>>>>>>>> Til: "programming at jsoftware.com" <programming at jsoftware.com>
>>>>>>>>>> Sendt: 0:40 tirsdag den 17. september 2013
>>>>>>>>>> Emne: Re: [Jprogramming] Plotting complex lists
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> Summary of results.  The strategy of hyperbola below (plotting a
>>>>>>>>>> complex
>>>>>>>>> table) is not well known.  Henry Rich found it and reported it.
>>>>>>>>>> 
>>>>>>>>>> Bo Jacoby gave the best way to change the sign of the real part of a
>>>>>>>>> complex number.
>>>>>>>>>> Simply do [: + -   .
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> NB. Complex Analytic Geometry
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> NB. How to calculate complex number lists and tables for NB. plotting
>>>>>>>>>> lines, circles, ellipses, and hyperbolas.  How to NB. modify these
>>>>>>>>>> tables to achieve translations, rotations, NB. and reflections.  Begin
>>>>>>>>>> with preliminaries:
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> steps =: {.@] + -~/@] * [ %~ [: i. >:@[
>>>>>>>>>> 
>>>>>>>>>> NB.  n steps a,b produces n+1 equally spaced values from a to b
>>>>>>>>>> 
>>>>>>>>>> to =: 512 steps ,  NB. Usage a to b for 512 steps from a to b
>>>>>>>>>> 
>>>>>>>>>> sin =: 1&o.
>>>>>>>>>> 
>>>>>>>>>> cos =: 2&o.
>>>>>>>>>> 
>>>>>>>>>> sinh =: 5&o.
>>>>>>>>>> 
>>>>>>>>>> cosh =: 6&o.
>>>>>>>>>> 
>>>>>>>>>> arcsinh =: _5&o.
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> NB. Now, results
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> line =: 2 : 'm + (n-m)*]'
>>>>>>>>>> 
>>>>>>>>>> NB. A line B [ t is point "t of the way from A to B".  Command NB.
>>>>>>>>>> NB.    plot 0 line 1j1 [ _1 to 2
>>>>>>>>>> NB.
>>>>>>>>>> NB. shows the line segment from _1j_1 to 2j2
>>>>>>>>>> 
>>>>>>>>>> NB. You are plotting a list of 513 complex numbers.
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> parabola =: 1 : '] j. (1 % 4 * m) * *:'
>>>>>>>>>> 
>>>>>>>>>> NB. p parabola x produces point x j. y on parabola NB. (*: x) = 4*p*y
>>>>>>>>>> . Command NB.
>>>>>>>>>> NB.    plot 1r4 parabola _2 to 2
>>>>>>>>>> NB.
>>>>>>>>>> NB. plots parabola y = *: x for x from _2 to 2
>>>>>>>>>> 
>>>>>>>>>> NB. You are plotting a list of 513 complex numbers.
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> ellipse =: 2 : '((m * cos) j. n * sin) 0 to 2p1'
>>>>>>>>>> 
>>>>>>>>>> NB. Suggested by Henry Rich
>>>>>>>>>> 
>>>>>>>>>> NB. Command
>>>>>>>>>> NB.
>>>>>>>>>> NB.    plot a ellipse b
>>>>>>>>>> NB.
>>>>>>>>>> NB. plots the ellipse 1 = (*: x % a) + *: y % b .
>>>>>>>>>> 
>>>>>>>>>> NB. If a = b you get the circle (*: x) + (*: y) = *: a
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> hyperbola =: 2 : '[: (,: +) (n * sinh) j. m * cosh'
>>>>>>>>>> 
>>>>>>>>>> NB. Suggested by Henry Rich
>>>>>>>>>> 
>>>>>>>>>> toh =: [: to/ [: arcsinh %~
>>>>>>>>>> 
>>>>>>>>>> NB. b toh c,d is (arcsinh c%b) to (arcsinh d%b)
>>>>>>>>>> 
>>>>>>>>>> NB. Command
>>>>>>>>>> NB.
>>>>>>>>>> NB.    plot a hyperbola b [ b toh c,d
>>>>>>>>>> NB.
>>>>>>>>>> NB. plots  y^2/a^2 - x^2/b^2 = 1  for x from c to d.
>>>>>>>>>> 
>>>>>>>>>> NB. Remember the pattern b [ b toh c,d
>>>>>>>>>> 
>>>>>>>>>> NB. You are plotting rows of a 2 by 513 table to get the two NB.
>>>>>>>>>> branches of the hyperbola.
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> NB. Rotations, translations, and reflections
>>>>>>>>>> 
>>>>>>>>>> NB. Multiply a complex number list or table by (^&j. theta) NB. to
>>>>>>>>>> rotate all of its points by theta radians.  The center NB. of rotation
>>>>>>>>>> is the origin 0 = 0j0 .
>>>>>>>>>> 
>>>>>>>>>> NB. Add 5j3 to a complex list or table to move all of its points NB.
>>>>>>>>>> the distance and direction of 5j3 from 0j0.
>>>>>>>>>> 
>>>>>>>>>> NB. Use (+ list) or (+ table) (monadic + is conjugate) to NB. reflect
>>>>>>>>>> all the points of the list or table across the NB. line through 0j0
>>>>>>>>>> and
>>>>>>>>>> 1j0 -- the x-axis.  Afterwards NB. multiply by (^&j. theta) to achieve
>>>>>>>>>> a reflection across NB. the line through 0j0 and (^&j. theta).
>>>>>>>>>> 
>>>>>>>>>> NB. Multiply a positive number p times a list or table to NB. achieve
>>>>>>>>>> an expansion from 0 or compression toward 0 NB. according as  p > 1
>>>>>>>>>> or p < 1  .
>>>>>>>>>> 
>>>>>>>>>> NB. If you want to combine several operations do the NB. reflection
>>>>>>>>>> first and the translation last.
>>>>>>>>>> 
>>>>>>>>>> NB. Example
>>>>>>>>>> NB.
>>>>>>>>>> NB.    plot (^&j. theta) * p parabola _2 to 3 NB.
>>>>>>>>>> NB. plots a parabola rotated by theta radians, with 0j0 NB. the center
>>>>>>>>>> of rotation.  If theta is _1r2p1 (that is NB. - pi%2 radians) you have
>>>>>>>>>> converted a  (*: x) = 4 * p * y NB. parabola into a  (*: y) = 4 * p *
>>>>>>>>>> x parabola.
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> --Kip Murray
>>>>>>>>>> 
>>>>>>>>>> Sent from my iPad
>>>>>>>>>> 
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