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equation solving - Finding intersections of two parametric curves


I am dealing with the root finding of two parametric data curves (data below).


I have already checked Parametric Interpolation of 2D data, which I found useful for plotting, though it is not adequate for interpolating the data and consequently, for finding the point/s where both curves cross.


dat=# {1,10}&/@SortBy[im,-#[[1]]&];
a=First@dat;
b=Rest@dat;
pathim={1,.1} #&/@First@Last@Reap@Do[b=DeleteCases[b,a=Sow@Nearest[b,a][[1]],1];,{Length@b}];

dat=# {1,10}&/@SortBy[re,-#[[1]]&];
a=First@dat;
b=Rest@dat;
pathre={1,.1} #&/@First@Last@Reap@Do[b=DeleteCases[b,a=Sow@Nearest[b,a][[1]],1];,{Length@b}];
Graphics[Line@{pathre,pathim},AspectRatio->1,Frame->True]

Crossing of parametric curves


Consider that if you perform a parametric interpolation, you would still have the problem that both curves will have different parameters $t$ and $t'$.


Any help will be welcome. Thanks in advance,


Francisco



im={{0.0531041079509168`,0.4537856055185257`},{0.05466627362026595`,0.4363323129985824`},{0.055371695336473604`,0.47123889803846897`},{0.0587862063093332`,0.4188790204786391`},{0.05966729343396994`,0.4886921905584123`},{0.06002132511085396`,0.415948661223835`},{0.06002132511085396`,0.4896587722299967`},{0.06575305305020286`,0.4014257279586958`},{0.06620599426144534`,0.5061454830783556`},{0.07002487929599628`,0.3948582697541205`},{0.07002487929599628`,0.5135624010553028`},{0.07512591694116288`,0.5235987755982988`},{0.07616913318690677`,0.3839724354387525`},{0.08002843348113861`,0.37928209464467316`},{0.08002843348113861`,0.5333666581380405`},{0.08446817154140783`,0.5410520681182421`},{0.09003198766628094`,0.36758426429054275`},{0.09003198766628094`,0.5502658471113652`},{0.09086816581576561`,0.3665191429188092`},{0.09506524408809112`,0.5585053606381855`},{0.10003554185142327`,0.3585188972184474`},{0.10003554185142327`,0.5668389658408651`},{0.10599987363255793`,0.5759586531581288`},{0.11003909603656559`,0.3499821309207754`},{0.11003909603656559`,0.5822588977132048`},{0.11118556664133165`,0.3490658503988659`},{0.11751340259454512`,0.5934119456780721`},{0.12004265022170792`,0.344202114592412`},{0.12004265022170792`,0.5973515337967324`},{0.12869769426292388`,0.6108652381980153`},{0.13004620440685025`,0.33895717609139303`},{0.13004620440685025`,0.6131207072459866`},{0.13943871850933623`,0.6283185307179586`},{0.14004975859199256`,0.3337693543200807`},{0.14004975859199256`,0.6293735517037298`},{0.1448928765578632`,0.33161255787892263`},{0.15001734189017396`,0.6457718232379019`},{0.1500533127771349`,0.3299538513479695`},{0.1500533127771349`,0.6458374688532782`},{0.16005686696227722`,0.32724984607051905`},{0.16005686696227722`,0.663160407666875`},{0.1600947333741542`,0.6632251157578453`},{0.16939105761415585`,0.6806784082777885`},{0.17006042114741954`,0.32510426347930604`},{0.17006042114741954`,0.6820027152855632`},{0.17834315993959135`,0.6981317007977318`},{0.18006397533256188`,0.323456717685147`},{0.18006397533256188`,0.7020915043821209`},{0.18659140121444864`,0.7155849933176751`},{0.1900675295177042`,0.3224938152915452`},{0.1900675295177042`,0.7234506689272742`},{0.19432724424416561`,0.7330382858376184`},{0.20007108370284654`,0.32194010589163236`},{0.20007108370284654`,0.7473057900969771`},{0.20140343554157758`,0.7504915783575618`},{0.20824586759352762`,0.767944870877505`},{0.21007463788798886`,0.32198227941312596`},{0.21007463788798886`,0.7735544586917432`},{0.21440848678980026`,0.7853981633974483`},{0.22007819207313117`,0.32268480961476603`},{0.23008174625827352`,0.323717711816065`},{0.24008530044341583`,0.32536509061435503`},{0.2500888546285582`,0.32760449250820817`},{0.2600924088137005`,0.33028549462451157`},{0.26430223561363364`,0.33161255787892263`},{0.2700959629988428`,0.3339970032425843`},{0.2800995171839851`,0.33833635585161864`},{0.29010307136912744`,0.3431145125802778`},{0.3001066255542698`,0.348219551120407`},{0.3016794563236875`,0.3490658503988659`},{0.31011017973941213`,0.35487194552795387`},{0.32011373392455444`,0.36175350208401263`},{0.32674322227218133`,0.3665191429188092`},{0.33011728810969676`,0.3694992086937793`},{0.3401208422948391`,0.3779490778028813`},{0.34724156791654076`,0.3839724354387525`},{0.35012439647998145`,0.3869800216501857`},{0.36012795066512376`,0.397301449763025`},{0.3642757416734221`,0.4014257279586958`},{0.3701315048502661`,0.40818604597717256`},{0.3798968075860652`,0.4188790204786391`},{0.3801350590354084`,0.4191811250284763`},{0.3901386132205507`,0.4316677521878405`},{0.39414905060506983`,0.4363323129985824`},{0.4001421674056931`,0.44396444613816133`},{0.4080491550187325`,0.4537856055185257`},{0.4101457215908354`,0.45676212790254306`},{0.4201492757759777`,0.4696249603539099`},{0.42149486869254693`,0.47123889803846897`},{0.43015282996112003`,0.48258796657484376`},{0.4350833036679111`,0.4886921905584123`},{0.44015638414626235`,0.4954819929471901`},{0.4487072833146444`,0.5061454830783556`},{0.45015993833140466`,0.508055663432461`},{0.46016349251654703`,0.5202514638628473`},{0.46307521883939134`,0.5235987755982988`},{0.47016704670168935`,0.5322780570272747`},{0.477914190905029`,0.5410520681182421`},{0.48017060088683167`,0.5437317838666962`},{0.490174155071974`,0.5544902541741973`},{0.4941879210211103`,0.5585053606381855`},{0.5001777092571164`,0.5648273949103115`},{0.5101812634422587`,0.5748053095034658`},{0.5114599742898591`,0.5759586531581288`},{0.520184817627401`,0.5841495962231561`},{0.5301883718125433`,0.5931600434577045`},{0.5304986382227892`,0.5934119456780721`},{0.5401919259976856`,0.6015014405876375`},{0.5501954801828279`,0.6092182031363506`},{0.5525479998123303`,0.6108652381980153`},{0.5601990343679702`,0.6165152214659495`},{0.5702025885531126`,0.6233328283352525`},{0.5781515740734545`,0.6283185307179586`},{0.5802061427382549`,0.6296561108077645`},{0.5902096969233973`,0.6355223762629595`},{0.6002132511085396`,0.6408151697372594`},{0.6102168052936819`,0.6457035790124366`},{0.6103667848233747`,0.6457718232379019`},{0.6202203594788243`,0.6504250134981324`},{0.6302239136639666`,0.6543542727813271`},{0.6402274678491089`,0.6577018258959079`},{0.6502310220342512`,0.6608382810360297`},{0.65893148745905`,0.6632251157578453`},{0.6602345762193935`,0.6636086146790213`},{0.6702381304045358`,0.6660358970840763`},{0.6802416845896782`,0.6679454676068672`},{0.6902452387748205`,0.6692245391152624`}};re={{0.3766619334176538`,0.6283185307179586`},{0.3774833614565732`,0.6457718232379019`},{0.37836289304330173`,0.5934119456780721`},{0.3794436621915606`,0.6632251157578453`},{0.3800679818708404`,0.5844653112819048`},{0.3800679818708404`,0.667375571838723`},{0.3818120099506517`,0.5759586531581288`},{0.38212616179315456`,0.6806784082777885`},{0.38540217552140743`,0.6981317007977318`},{0.3882567605401741`,0.5585053606381855`},{0.38938774795241116`,0.7155849933176751`},{0.39006977086744143`,0.5551487025539988`},{0.39006977086744143`,0.7184208526690175`},{0.39384490677070083`,0.7330382858376184`},{0.3983398475296372`,0.5410520681182421`},{0.3990895276046064`,0.7504915783575618`},{0.4000715598640425`,0.5390451811857366`},{0.4000715598640425`,0.7536360014898048`},{0.40490154189174227`,0.767944870877505`},{0.4100733488606436`,0.5288508576882919`},{0.4100733488606436`,0.7829547200266151`},{0.4109671363156394`,0.7853981633974483`},{0.41590869326136504`,0.5235987755982988`},{0.42007513785724465`,0.5208888749031353`},{0.4300769268538457`,0.5152293488576334`},{0.4400787158504468`,0.5105095807330549`},{0.45008050484704787`,0.5063401987861282`},{0.45064664649866976`,0.5061454830783556`},{0.4600822938436489`,0.5036814650968815`},{0.47008408284024994`,0.5014626165676501`},{0.48008587183685103`,0.4996348462112899`},{0.4900876608334521`,0.4982219621388478`},{0.5000894498300532`,0.49707951766544517`},{0.5100912388266542`,0.4964390647155012`},{0.5200930278232553`,0.4958486047744673`},{0.5300948168198564`,0.49545500725491304`},{0.5400966058164574`,0.49521155256588884`},{0.5500983948130584`,0.49514685900067146`},{0.5601001838096595`,0.4952291466547238`},{0.5701019728062606`,0.4952990199882136`},{0.5801037618028617`,0.4954981261098205`},{0.5901055507994627`,0.49581927136967574`},{0.6001073397960638`,0.4961027882281859`},{0.6101091287926649`,0.4963622147030731`},{0.6201109177892659`,0.4966042118151429`},{0.630112706785867`,0.49683858164676903`},{0.640114495782468`,0.49705190249915443`},{0.6501162847790691`,0.49712592063118116`},{0.6601180737756702`,0.49718413826216856`},{0.6701198627722712`,0.4972295004992033`},{0.6801216517688723`,0.4972077101740653`},{0.6901234407654734`,0.4970539255431156`}};

Answer



You've already done the hard work in putting the path together. Now you could just do a quick and dirty interpolation and FindRoot.


rex = Interpolation[Thread[{Range@Length@pathre, First /@ pathre}]];
rey = Interpolation[Thread[{Range@Length@pathre, Last /@ pathre}]];
imx = Interpolation[Thread[{Range@Length@pathim, First /@ pathim}]];
imy = Interpolation[Thread[{Range@Length@pathim, Last /@ pathim}]];

{rex[t], rey[t]} /. FindRoot[{rex[t] - imx[s], rey[t] - imy[s]}, {{s, 1}, {t, 1}}]



{0.436125,0.512288}



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