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Re: Ploting a transformation of a set

• To: mathgroup at smc.vnet.net
• Subject: [mg123474] Re: Ploting a transformation of a set
• From: Barrie Stokes <Barrie.Stokes at newcastle.edu.au>
• Date: Thu, 8 Dec 2011 05:26:16 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <4EDCB5F0.813B.006A.0@newcastle.edu.au> <jbkj0l$i30$1@smc.vnet.net>

Hi Dan

This is very nice too.

I think stabilising the mapped plane with PlotRange helps:

With[ {n = 10, c1 = ColorData[ 1 ][ 1 ], c2 = ColorData[ 1 ][ 2 ]},
   refGrid = Show[ {
        ParametricPlot[ Table[ {2 t, 2 (k/n)}, {k, n} ], {t, 0, 1}, 
     PlotStyle -> c1 ],
        ParametricPlot[ Table[ {2 (k/n), 2 t}, {k, n} ], {t, 0, 1}, 
     PlotStyle -> c2 ]
      }, ImageSize -> 200 ];
   Manipulate[ 
      Module[ {map},
         map = ({x, y} \[Function] {(x + a y)^b, (a x + y)^b});
         Grid[ {{
              refGrid,
              Show[ {                  
        ParametricPlot[ Table[ map @@ {2 t, 2 (k/n)}, {k, n} ], {t,
          0, 1}, PlotStyle -> c1, PlotRange -> {{0, 2}, {0, 2}} ],
                  
        ParametricPlot[ Table[ map @@ {2 (k/n), 2 t}, {k, n} ], {t,
          0, 1}, PlotStyle -> c2, PlotRange -> {{0, 2}, {0, 2}} ]
                }, ImageSize -> 200 ]
           }} ]
    ],
    {{a, 0.5}, 0, 1, 0.05},
    {{b, 0.5}, 0.1, 1, 0.05}
   ]
  ]

Cheers

Barrie

PS
I like your indenting for clarity. I purposely use "[space" and "space]" everywhere in code I paste into emails. Gradually my spellchecker will know every reserved word in Mathematica.

>>> On 07/12/2011 at 10:15 pm, in message <201112071115.GAA04280 at smc.vnet.net>, Dan
<dflatin at rcn.com> wrote:
> Another approach would be to look at the transformation of the grid. I
> find this sort of visualization more informative for these sorts of
> mappings. For example:
> 
> With[{n=10,c1=ColorData[1][1],c2=ColorData[1][2]},
>   refGrid=Show[{
>     ParametricPlot[Table[{2t,2(k/n)},{k,n}],{t,0,1},PlotStyle->c1],
>     ParametricPlot[Table[{2(k/n),2t},{k,n}],{t,0,1},PlotStyle->c2]
>   },ImageSize->200];
>   Manipulate[
>     Module[{map},
>       map=({x,y}\[Function]{(x+a y)^b,(a x+y)^b});
>       Grid[{{
>         refGrid,
>         Show[{
>           ParametricPlot[Table[map@@{2t,2(k/n)},{k,n}],{t,
> 0,1},PlotStyle->c1],
>           ParametricPlot[Table[map@@{2(k/n),2t},{k,n}],{t,
> 0,1},PlotStyle->c2]
>         },ImageSize->200]
>      }}]
>   ],
>   {{a,0.5},0,1,0.05},
>   {{b,0.5},0.1,1,0.05}
>   ]
> ]
> 
> -- Dan