Re: Concentric contours about the centroid, having the same length, and interior to an initial contour.

• To: mathgroup at smc.vnet.net
• Subject: [mg43964] Re: Concentric contours about the centroid, having the same length, and interior to an initial contour.
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Thu, 16 Oct 2003 04:15:59 -0400 (EDT)
• Organization: Universitaet Leipzig
• References: <bmj2mv\$prl\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

you need a 2d level set method.

Construct a signed distance function psi[x,y]
from the contur (|psi[0,x,y]| is the distance to of the point (x,y) to
the contour and inner points have Sign[psi]<0 and outer points
Sign[psi]>0) for and look how the function envolve under

D[psi[t,x,y],t]= nabla[nabla[psi]/Abs[nabla[psi]]]

psi[t,x,y]==0 is than your familiy of new contours ..

Regards
Jens

Gilmar Rodr?guez Pierluissi wrote:
>
> I have a basic contour defined by:
>
> In[1]:
>
> contour = {{10545, 28012}, {10232, 28166}, {10027, 28218},
> {9714, 28115}, {9354, 28166}, {9042, 28115}, {8785, 28064},
> {8575, 27858}, {8160, 27858}, {7954, 27546}, {8108, 27238},
> {8421, 27238}, {8836, 27079}, {9042, 26874}, {9247, 26510},
> {9406, 26197}, {9611, 25992}, {9662, 25627}, {9560, 25268},
> {9196, 25007}, {8939, 24750}, {8678, 24750}, {8421, 24386},
> {8216, 24125}, {8160, 23919}, {8006, 23607}, {8006, 23401},
> {7852, 23037}, {7749, 22832}, {7539, 22678}, {7179, 22678},
> {6918, 22570}, {6713, 22468}, {6451, 22365}, {6036, 22365},
> {5728, 22468}, {5677, 22104}, {5625, 21693}, {5625, 21483},
> {5364, 21483}, {5159, 21950}, {5000, 22211}, {4846, 22519},
> {4538, 22570}, {4641, 22160}, {4949, 21950}, {4949, 21642},
> {5159, 21278}, {5313, 21068}, {5569, 20862}, {5569, 20708},
> {5518, 20447}, {5415, 20190}, {5415, 19929}, {5313, 19565},
> {5000, 19308}, {4743, 19308}, {4589, 19411}, {4431, 19565},
> {4225, 19672}, {4020, 19621}, {3758, 19621}, {3758, 19359},
> {4020, 19154}, {4328, 18944}, {4482, 18585}, {4897, 18739},
> {5210, 18841}, {5364, 18426}, {5467, 18118}, {5625, 17754},
> {5831, 17339}, {5625, 17026}, {5364, 16821}, {5056, 16667},
> {4897, 16200}, {5000, 15887}, {5159, 15579}, {5107, 15318},
> {4897, 15005}, {4538, 15005}, {4174, 15215}, {4020, 15420},
> {4071, 15112}, {4277, 14851}, {4589, 14697}, {4897, 14594},
> {4897, 14179}, {4897, 13866}, {4743, 13507}, {4795, 13143},
> {4795, 12830}, {4692, 12363}, {4589, 12107}, {4589, 11794},
> {4328, 11486}, {3964, 11327}, {3861, 10968}, {4020, 10604},
> {3861, 10086}, {3810, 9670}, {3707, 9255}, {3446, 8737},
> {3292, 8219}, {3343, 7804}, {3502, 7337}, {3189, 6977},
> {2928, 6926}, {2620, 6870}, {2466, 6613}, {2256, 6147},
> {2050, 6352}, {1948, 6613}, {1686, 6767}, {1481, 6716},
> {1117, 6716}, {804, 6926}, {650, 6819}, {547, 6562},
> {445, 6352}, {393, 5993}, {235, 5941}, {81, 5629},
> {286, 5475}, {701, 5423}, {911, 5372}, {963, 5111},
> {860, 4905}, {650, 4695}, {650, 4280}, {1014, 4126},
> {1271, 3869}, {1686, 3659}, {2050, 3505}, {2466, 3402},
> {2774, 3351}, {3086, 3454}, {3446, 3659}, {3861, 3710},
> {4174, 3659}, {4589, 3710}, {5056, 3608}, {5313, 3244},
> {5467, 2726}, {5677, 2469}, {5933, 2418}, {6195, 2208},
> {6451, 1900}, {6503, 1484}, {6713, 1171}, {6815, 812},
> {7179, 653}, {7488, 551}, {7800, 345}, {8057, 551},
> {8267, 397}, {8575, 187}, {8836, 33}, {8836, 238},
> {8626, 499}, {8524, 705}, {8472, 966}, {8626, 1171},
> {8267, 1274}, {8216, 1069}, {8057, 966}, {7800, 1069},
> {7539, 1171}, {7385, 1330}, {7282, 1690}, {7590, 1741},
> {7642, 1900}, {7642, 2208}, {7800, 2469}, {7590, 2674},
> {7282, 2572}, {7021, 2418}, {6815, 2418}, {6815, 2833},
> {6918, 3038}, {7123, 3090}, {6918, 3244}, {6713, 3454},
> {6605, 3556}, {6451, 3869}, {6713, 3972}, {7021, 4126},
> {7385, 4228}, {6918, 4331}, {6713, 4438}, {6451, 4592},
> {6195, 4228}, {5882, 4387}, {5728, 4695}, {5728, 5008},
> {5625, 5475}, {5728, 5890}, {6143, 5890}, {6661, 5890},
> {7072, 5941}, {6815, 6198}, {6400, 6198}, {6087, 6147},
> {6246, 6511}, {6451, 6819}, {6713, 7080}, {6969, 7495},
> {7231, 7495}, {7800, 7598}, {7436, 7752}, {7179, 7855},
> {7072, 8168}, {6764, 8480}, {6713, 9050}, {6918, 9465},
> {7123, 9824}, {7282, 10240}, {7590, 10968}, {8057, 11379},
> {8318, 11794}, {8785, 12312}, {8990, 12727}, {9406, 13091},
> {9457, 13661}, {9714, 14333}, {10027, 14748}, {10339, 15420},
> {10909, 15579}, {11632, 15733}, {11945, 15630}, {12355, 15318},
> {12981, 15215}, {13443, 14800}, {13653, 14333}, {13756, 13918},
> {13910, 13451}, {14222, 13091}, {14586, 12830}, {14689, 12415},
> {14638, 11948}, {14535, 11430}, {14274, 11071}, {13961, 10707},
> {13807, 10655}, {13602, 10394}, {13961, 10394}, {14274, 10553},
> {14428, 10342}, {14376, 9983}, {14535, 9773}, {14843, 9824},
> {14946, 9568}, {15104, 9358}, {15258, 9101}, {15361, 8788},
> {15566, 8840}, {15982, 8947}, {16243, 8891}, {16603, 8788},
> {16864, 8532}, {17279, 8373}, {17587, 8270}, {17849, 7911},
> {18208, 7649}, {18315, 7080}, {18469, 6767}, {18782, 6665},
> {18936, 6408}, {19193, 6044}, {19557, 5834}, {19972, 6044},
> {20178, 6352}, {20644, 6511}, {20747, 6870}, {20798, 7234},
> {20542, 7547}, {20388, 8014}, {20229, 8322}, {20075, 8634},
> {19921, 9152}, {20229, 9465}, {20490, 9722}, {20747, 9876},
> {20490, 9876}, {20178, 9824}, {19972, 9619}, {19660, 9619},
> {19608, 9876}, {19300, 10086}, {18885, 10137}, {18675, 10342},
> {18623, 10604}, {18782, 10809}, {18936, 11019}, {18572, 11173},
> {18469, 10912}, {18315, 10655}, {18208, 10291}, {17900, 10394},
> {17587, 10394}, {17177, 10342}, {16761, 10394}, {16551, 10450},
> {16295, 10604}, {15879, 10707}, {15566, 11019}, {15622, 11379},
> {15828, 11794}, {16033, 12107}, {15725, 12522}, {15622, 12779},
> {15515, 13143}, {15776, 13451}, {15776, 13764}, {15464, 14025},
> {15258, 14333}, {15048, 14543}, {14894, 14748}, {14843, 14902},
> {14689, 15164}, {14689, 15420}, {14586, 15836}, {14376, 16200},
> {14222, 16615}, {14120, 16872}, {13910, 16923}, {13756, 17026},
> {13499, 17236}, {13186, 17236}, {12981, 17390}, {12668, 17493},
> {12304, 17651}, {12047, 17703}, {11786, 17600}, {11427, 17544},
> {11011, 17600}, {10493, 17600}, {9924, 17651}, {9611, 17908},
> {9457, 18169}, {9714, 18585}, {10181, 18944}, {10391, 19462},
> {10339, 19878}, {10232, 20293}, {10181, 20657}, {10699, 20862},
> {11063, 21278}, {11375, 21847}, {11735, 22160}, {12201, 22468},
> {12514, 22314}, {12719, 21898}, {13032, 21586}, {13084, 22052},
> {13238, 22314}, {13238, 22519}, {12873, 22570}, {12719, 22883},
> {12719, 23350}, {13032, 23765}, {12925, 24125}, {12719, 24437},
> {12463, 24801}, {12355, 25058}, {12047, 25525}, {11837, 25940},
> {11632, 26253}, {11319, 26253}, {11011, 26253}, {10750, 26458},
> {10596, 26822}, {10596, 27182}, {10596, 27494}, {10545, 28012}};
>
> The length of the contour is given by:
>
> In[2]:
>
> Length[contour]
>
> Out[2]:
>
> 375
>
> A plot of this contour is given by:
>
> In[3]:
>
> plt1 = ListPlot[contour, PlotStyle -> {Hue[0], PointSize[.01]}]
>
> Out[3]: graphics
>
> The centroid of the contour is given by:
>
> In[4]:
>
> centroid = N[Plus @@ contour/Length[contour]]
>
> Out[4]: {9426.41, 12877.6}
>
> My question is:
>
> How can I generate (say) five contours, so that:
>
> (1.) Each contour (set) have the same length as the original contour
>
>  (in this case, each contour should contain  375 {x,y} points).
>
> (2.) Each contour is equidistant, and concentric (with respect to
>
>  the centroid) to the previous contour.
>
> (3.) The five contours are inside the original contour defined above?
>
> Thank you for your help!

```

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