Border Curves
Border Curves
In[107]:=
points={{0.105085, 0.311864}, {0.125424, 0.308475},
{0.142373, 0.305085}, {0.147458, 0.301695},
{0.154237, 0.29661}, {0.166102, 0.289831},
{0.177966, 0.283051}, {0.189831, 0.276271},
{0.201695, 0.269492}, {0.210169, 0.264407},
{0.222034, 0.262712}, {0.237288, 0.259322},
{0.250847, 0.255932}, {0.264407, 0.254237},
{0.267797, 0.259322}, {0.271186, 0.264407},
{0.281356, 0.261017}, {0.291525, 0.261017},
{0.30339, 0.259322}, {0.310169, 0.250847},
{0.316949, 0.240678}, {0.327119, 0.233898},
{0.332203, 0.222034}, {0.337288, 0.208475},
{0.345763, 0.2}, {0.354237, 0.194915},
{0.362712, 0.189831}, {0.366102, 0.2},
{0.376271, 0.208475}, {0.386441, 0.208475},
{0.39661, 0.205085}, {0.40339, 0.19661},
{0.410169, 0.186441}, {0.416949, 0.172881},
{0.425424, 0.162712}, {0.428814, 0.150847},
{0.432203, 0.140678}, {0.438983, 0.130508},
{0.450847, 0.125424}, {0.462712, 0.122034},
{0.471186, 0.118644}};
In[108]:=
ListPlot[points, AspectRatio->Automatic];
In[109]:=
usmexborder=SplineFit[points,Cubic]
Out[109]=
SplineFunction[Cubic,{0., 40.},<>]
In[110]:=
ParametricPlot[usmexborder[x],{x,0,14},PlotRange->All,Compiled->False];
You can evaluate the function usmexborder at any point along the border. Remember that the spline curve is a parametric curve; you you enter a parameter within the range of the parametrization and the output is the (x,y) pair at that parameter.
In[111]:=
usmexborder[0]
usmexborder[14]
Out[111]=
{0.105085, 0.311864}
Out[112]=
{0.267797, 0.259322}
The following command gives the type of spline (Cubic), the points used, and the coefficients of the polynomials that make up the spline (if you really want to know the function). See the next example.
In[113]:=
InputForm[usmexborder]
Out[113]=
SplineFunction[Cubic, {0., 40.},
{{0.105085, 0.311864}, {0.125424, 0.308475},
{0.142373, 0.305085}, {0.147458, 0.301695},
{0.154237, 0.29661}, {0.166102, 0.289831},
{0.177966, 0.283051}, {0.189831, 0.276271},
{0.201695, 0.269492}, {0.210169, 0.264407},
{0.222034, 0.262712}, {0.237288, 0.259322},
{0.250847, 0.255932}, {0.264407, 0.254237},
{0.267797, 0.259322}, {0.271186, 0.264407},
{0.281356, 0.261017}, {0.291525, 0.261017},
{0.30339, 0.259322}, {0.310169, 0.250847},
{0.316949, 0.240678}, {0.327119, 0.233898},
{0.332203, 0.222034}, {0.337288, 0.208475},
{0.345763, 0.2}, {0.354237, 0.194915},
{0.362712, 0.189831}, {0.366102, 0.2},
{0.376271, 0.208475}, {0.386441, 0.208475},
{0.39661, 0.205085}, {0.40339, 0.19661},
{0.410169, 0.186441}, {0.416949, 0.172881},
{0.425424, 0.162712}, {0.428814, 0.150847},
{0.432203, 0.140678}, {0.438983, 0.130508},
{0.450847, 0.125424}, {0.462712, 0.122034},
{0.471186, 0.118644}},
{{{0.105085, 0.02038907441381466968, 0.,
-0.00005007441381466968189},
{0.311864, -0.003364837091438011499, 0.,
-0.0000241629085619885052}},
{{0.125424, 0.02023885117237066064,
-0.0001502232414440090406, -0.003139627930926651602}\
, {0.308475, -0.003437325817123977015,
-0.00007248872568596551645, 0.0001198145428099425567}
}, {{0.142373, 0.01051952089670268775,
-0.009569107034223963847, 0.004134586137521276099},
{0.305085, -0.003222859640066080377,
0.0002869549027438621538, -0.000454095262677781778}},
{{0.147458, 0.003785065240818588354,
0.002834651378339864452, 0.0001592833808415471856},
{0.301695, -0.004011235622611701403,
-0.00107533088528948318, 1.566507901184580257*10^-6}}\
, {{0.154237, 0.009932218140022958815,
0.00331250152086450601, -0.001379719660887464821},
{0.29661, -0.00615719786948711402,
-0.001070631361585929435, 0.0004488292310730434494}},
{{0.166102, 0.01241806219908957637,
-0.0008266574617978884527, 0.0002725952627083120887}\
, {0.289831, -0.006951972899439842541,
0.0002758563316332009146, -0.0001038834321933583753}}\
, {{0.177966, 0.01158253306361873573,
-(8.871673672952190723*10^-6),
0.0002913386100542164513},
{0.283051, -0.006711910532753515839,
-0.0000357939649468742163,
-0.00003229550229960994776}},
{{0.189831, 0.0124388055464354807,
0.0008651441564896971631, -0.001439949702925177859},
{0.276271, -0.006880384969546094113,
-0.0001326804718457040554, 0.0002340654413917981627}}\
, {{0.201695, 0.009849244750639341453,
-0.003454704952285836412, 0.002079460201646494952},
{0.269492, -0.006443549589062107735,
0.0005695158523296904336, 0.0007890337367324172998}},
{{0.210169, 0.009178215451007153484,
0.002783675652653648439, -0.00009689110366080192068}\
, {0.264407, -0.002937416674205474969,
0.002936617062526942332, -0.001694200388321467363}},
{{0.222034, 0.0144548934453320446,
0.002493002341671242678, -0.001693895787003287289},
{0.262712, -0.002146783714115992395,
-0.002145984102437459758, 0.0009027678165534521514}},
{{0.237288, 0.0143592107676646681,
-0.002588685019338619183, 0.001788474251673951082},
{0.259322, -0.003730448469330555456,
0.0005623193472228966979, -0.0002218708778923412435}}\
, {{0.250847, 0.01454726348400928297,
0.00277673773568323406, -0.00376400121969251703},
{0.255932, -0.00327142240856178579,
-0.0001032932864541270323, 0.001679715695015912822}},
{{0.264407, 0.008808735296298200008,
-0.008515265923394317025, 0.003096530627096117017},
{0.254237, 0.001561138103577698612,
0.004935853798593611434, -0.001411991902171310044}},
{{0.267797, 0.001067795330797917011,
0.000774325957894034027, 0.001546878711308048966},
{0.259322, 0.007196869994250991348,
0.0006998780920796813014, -0.002811748086330672648}},
{{0.271186, 0.007257083380510131963,
0.005414962091818180927, -0.002502045472328312885},
{0.264407, 0.0001613819194183360074,
-0.007735366166912336643, 0.004183984247494000633}},
{{0.281356, 0.01058087114716155516,
-0.002091174325166757726, 0.001679303178005202574},
{0.261017, -0.002757397671924335376,
0.004816586575569665259, -0.002059188903645329883}},
{{0.291525, 0.01143643203084364743,
0.002946735208848849997, -0.00251816723969249742},
{0.261017, 0.0006982087682790054928,
-0.001360980135366324391, -0.001032228632912681103}},
{{0.30339, 0.00977540072946385516,
-0.00460776651022864226, 0.001611365780764787078},
{0.259322, -0.005120437401191686597,
-0.004457666034104367697, 0.001103103435296054291}},
{{0.310169, 0.005393965051300931873,
0.0002263308320657189747, 0.001159704116633349155},
{0.250847, -0.01072645916351225912,
-0.001148355728216204823, 0.001705814891728463949}},
{{0.316949, 0.009325739065332417287,
0.003705443181965766439, -0.002861182247298183723},
{0.240678, -0.00790572594475927692,
0.003969088946969187023, -0.002843363002209910091}},
{{0.327119, 0.008153078687369398998,
-0.004878103559928784728, 0.001809024872559385736},
{0.233898, -0.00849763705745063315,
-0.004561000059660543256, 0.001194637117111176399}},
{{0.332203, 0.003823946185189986747,
0.0005489710577493724781, 0.0007120827570606407761},
{0.222034, -0.01403572582543819047,
-0.000977088708327014051, 0.001453814533765204522}},
{{0.337288, 0.007058136571870654032,
0.002685219328931294807, -0.001268355900801948836},
{0.208475, -0.011628459640796605,
0.003384354892968599514, -0.0002308952521719945159}},
{{0.345763, 0.008623507527327397138,
-0.001119848373474551702, 0.0009703408461471545714},
{0.2, -0.005552435611375389522,
0.002691669136452615966, -0.002224233525077226445}},
{{0.354237, 0.009294833318819757446,
0.00179117416496691201, -0.002611007483786669478},
{0.194915, -0.006841797913701836927,
-0.003981031438779063368, 0.00573882935248090029}},
{{0.362712, 0.005044159197393573028,
-0.006041848286393096425, 0.0043876890889995234},
{0.189831, 0.002412627266182737208,
0.01323545661866363751, -0.005479083884846374706}},
{{0.366102, 0.006123529891605950374,
0.007121218980605473768, -0.003075748872211424137},
{0.2, 0.0124462888489708881,
-0.003201795035875486612, -0.000769493813095401484}},
{{0.376271, 0.0111387212361826255,
-0.00210602763602879864, 0.001137306399846173144},
{0.208475, 0.003734217337933710423,
-0.005510276475161691063, 0.00177605913722798064}},
{{0.386441, 0.01033858516366354765,
0.00130589156350972079, -0.001475476727173268433},
{0.208475, -0.001958158200705729784,
-0.0001820990634777491446, -0.001249742735816521074}}\
, {{0.39661, 0.008523938109163183931,
-0.003120538618010084511, 0.001376600508846900583},
{0.205085, -0.006071584535110791292,
-0.003931327270927312362, 0.001527911806038103652}},
{{0.40339, 0.006412662399683716657,
0.001009262908530617239, -0.0006429253082143338915},
{0.19661, -0.009350503658851105063,
0.0006524081471869985887, -0.001470904488335893533}},
{{0.410169, 0.006502412292101949462,
-0.0009195130161123844337, 0.001197100724010434947},
{0.186441, -0.01245840082948478848,
-0.003760305317820682007, 0.002658706147305470499}},
{{0.416949, 0.008254688431908485436,
0.002671789155918920413, -0.002451477587827405846},
{0.172881, -0.012002893023209741,
0.004215813124095729493, -0.002381920100885988499}},
{{0.425424, 0.006243833980264108724,
-0.004682643607563297124, 0.001828809627299188401},
{0.162712, -0.01071702707767624751,
-0.002929947178562236005, 0.001781974256238483512}},
{{0.428814, 0.002364975647035079679,
0.0008037852743342680788, 0.0002202390786306522462},
{0.150847, -0.01123099866608526898,
0.002415975590153214531, -0.001353976924067945539}},
{{0.432203, 0.004633263431595572575,
0.001464502510226224819, 0.0006822340581782026075},
{0.140678, -0.01046097825798267654,
-0.001645955182050622089, 0.001936933440033298626}},
{{0.438983, 0.009608970626582630037,
0.00351120468476083264, -0.001256175311343462669},
{0.130508, -0.007942088301984024842,
0.004164845138049273787, -0.001306756836065248937}},
{{0.450847, 0.01286285406207390731,
-0.0002573212492695553668,
-0.0007405328128043519372},
{0.125424, -0.003532668534081224078,
0.000244574629853526974, -0.0001019060957723029044}},
{{0.462712, 0.01012661312512174076,
-0.002478919687682611176, 0.0008263065625608703914},
{0.122034, -0.003349237561691078843,
-0.0000611436574633817378, 0.00002038121915446057926}}
}]
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