Cubic bezier curve length

<!DOCTYPE html>
<html>
<head>
<title>Cubic bezier curve length</title>
<style>
#figure1 {
  border: 1px solid black;
}

.draggable {
  cursor: move;
}

.control-point {
  opacity: 0.5;
  stroke: none;
  fill: red;
}

text {
  font-size: 12px;
}
</style>
</head>
<body>
<svg id="figure1" width="400" height="400">
  <path d="M120 160 35 200 220 260 220 40" stroke="black" fill="none" id="controlPath"/>
  <path d="M120 160C35 200 220 260 220 40" stroke="green" fill="none" id="curve"/>
  <circle cx="120" cy="160" r="8" id="p1" class="control-point draggable"/>
  <circle cx="35" cy="200" r="8" id="p2" class="control-point draggable"/>
  <circle cx="220" cy="260" r="8" id="p3" class="control-point draggable"/>
  <circle cx="220" cy="40" r="8" id="p4" class="control-point draggable"/>
  <text x="20" y="20" id="curveLength">curve length: 0</text>
  <text x="120" y="160" dx="10" id="t1">p1: 120/160</text>
  <text x="35" y="200" dx="10" id="t2">p2: 35/200</text>
  <text x="220" y="260" dx="10" id="t3">p3: 220/260</text>
  <text x="220" y="40" dx="10" id="t4">p4: 220/40</text>
</svg>
<script>
// This is a port to JavaScript from [Arc length | A Primer on Bézier Curves]( https://pomax.github.io/bezierinfo/#arclength )
// MIT license
var controlPath = document.getElementById('controlPath');
var curve = document.getElementById('curve');
var p1 = document.getElementById('p1');
var p2 = document.getElementById('p2');
var p3 = document.getElementById('p3');
var p4 = document.getElementById('p4');
var dragElem = null;

var svg = document.getElementById('figure1');
svg.addEventListener('mousemove', onMouseMove);
svg.addEventListener('mouseup', onMouseUp);

var elements = [p1, p2, p3, p4];
for (i = 0; i < elements.length; i++) {
  elements[i].addEventListener('mousedown', onMouseDown);
}

var mouseOffsetX, mouseOffsetY;

function getClientPointInSVG(ev) {
  var p, m;

p = svg.createSVGPoint();
  p.x = ev.clientX;
  p.y = ev.clientY;

m = dragElem.getScreenCTM();
  return p.matrixTransform(m.inverse());
}

function onMouseDown(ev) {
  var p;

dragElem = ev.target;
  p = getClientPointInSVG(ev);
  mouseOffsetX = p.x - dragElem.getAttribute('cx');
  mouseOffsetY = p.y - dragElem.getAttribute('cy');
}

function onMouseMove(ev) {
  var p, x, y, circleID, circleTextID, circleTextElem;

if (!dragElem) {
    return;
  }

p = getClientPointInSVG(ev);
  x = p.x - mouseOffsetX;
  y = p.y - mouseOffsetY;
  dragElem.setAttribute('cx', x);
  dragElem.setAttribute('cy', y);

controlPath.setAttribute('d',
    'M' + p1.getAttribute('cx') + ' ' + p1.getAttribute('cy') +
    ' ' + p2.getAttribute('cx') + ' ' + p2.getAttribute('cy') +
    ' ' + p3.getAttribute('cx') + ' ' + p3.getAttribute('cy') +
    ' ' + p4.getAttribute('cx') + ' ' + p4.getAttribute('cy'));
  curve.setAttribute('d',
    'M' + p1.getAttribute('cx') + ' ' + p1.getAttribute('cy') +
    'C' + p2.getAttribute('cx') + ' ' + p2.getAttribute('cy') +
    ' ' + p3.getAttribute('cx') + ' ' + p3.getAttribute('cy') +
    ' ' + p4.getAttribute('cx') + ' ' + p4.getAttribute('cy'));

circleID = dragElem.getAttribute('id');
  circleTextID = circleID.replace(/^p/, 't');
  circleTextElem = document.getElementById(circleTextID);
  circleTextElem.setAttribute('x', x);
  circleTextElem.setAttribute('y', y);
  circleTextElem.firstChild.data = circleID + ': ' + x + '/' + y;
  updateCurveLengthText();
}

function onMouseUp(ev) {
  dragElem = null;
}

function updateCurveLengthText() {
  var textElem = document.getElementById('curveLength');
  var xs = [+p1.getAttribute('cx'), +p2.getAttribute('cx'), +p3.getAttribute('cx'), +p4.getAttribute('cx')];
  var ys = [+p1.getAttribute('cy'), +p2.getAttribute('cy'), +p3.getAttribute('cy'), +p4.getAttribute('cy')];
  var arcLength = getArcLength(xs, ys);
  textElem.firstChild.data = 'curve length: ' + arcLength;
}

// Legendre-Gauss abscissae (xi values, defined at i=n as the roots of the nth order Legendre polynomial Pn(x))
var tValues = [
  [],
  [],
  [-0.5773502691896257645091487805019574556476,0.5773502691896257645091487805019574556476],
  [0,-0.7745966692414833770358530799564799221665,0.7745966692414833770358530799564799221665],
  [-0.3399810435848562648026657591032446872005,0.3399810435848562648026657591032446872005,-0.8611363115940525752239464888928095050957,0.8611363115940525752239464888928095050957],
  [0,-0.5384693101056830910363144207002088049672,0.5384693101056830910363144207002088049672,-0.9061798459386639927976268782993929651256,0.9061798459386639927976268782993929651256],
  [0.6612093864662645136613995950199053470064,-0.6612093864662645136613995950199053470064,-0.2386191860831969086305017216807119354186,0.2386191860831969086305017216807119354186,-0.9324695142031520278123015544939946091347,0.9324695142031520278123015544939946091347],
  [0, 0.4058451513773971669066064120769614633473,-0.4058451513773971669066064120769614633473,-0.7415311855993944398638647732807884070741,0.7415311855993944398638647732807884070741,-0.9491079123427585245261896840478512624007,0.9491079123427585245261896840478512624007],
  [-0.1834346424956498049394761423601839806667,0.1834346424956498049394761423601839806667,-0.5255324099163289858177390491892463490419,0.5255324099163289858177390491892463490419,-0.7966664774136267395915539364758304368371,0.7966664774136267395915539364758304368371,-0.9602898564975362316835608685694729904282,0.9602898564975362316835608685694729904282],
  [0,-0.8360311073266357942994297880697348765441,0.8360311073266357942994297880697348765441,-0.9681602395076260898355762029036728700494,0.9681602395076260898355762029036728700494,-0.3242534234038089290385380146433366085719,0.3242534234038089290385380146433366085719,-0.6133714327005903973087020393414741847857,0.6133714327005903973087020393414741847857],
  [-0.1488743389816312108848260011297199846175,0.1488743389816312108848260011297199846175,-0.4333953941292471907992659431657841622000,0.4333953941292471907992659431657841622000,-0.6794095682990244062343273651148735757692,0.6794095682990244062343273651148735757692,-0.8650633666889845107320966884234930485275,0.8650633666889845107320966884234930485275,-0.9739065285171717200779640120844520534282,0.9739065285171717200779640120844520534282],
  [0,-0.2695431559523449723315319854008615246796,0.2695431559523449723315319854008615246796,-0.5190961292068118159257256694586095544802,0.5190961292068118159257256694586095544802,-0.7301520055740493240934162520311534580496,0.7301520055740493240934162520311534580496,-0.8870625997680952990751577693039272666316,0.8870625997680952990751577693039272666316,-0.9782286581460569928039380011228573907714,0.9782286581460569928039380011228573907714],
  [-0.1252334085114689154724413694638531299833,0.1252334085114689154724413694638531299833,-0.3678314989981801937526915366437175612563,0.3678314989981801937526915366437175612563,-0.5873179542866174472967024189405342803690,0.5873179542866174472967024189405342803690,-0.7699026741943046870368938332128180759849,0.7699026741943046870368938332128180759849,-0.9041172563704748566784658661190961925375,0.9041172563704748566784658661190961925375,-0.9815606342467192506905490901492808229601,0.9815606342467192506905490901492808229601],
  [0,-0.2304583159551347940655281210979888352115,0.2304583159551347940655281210979888352115,-0.4484927510364468528779128521276398678019,0.4484927510364468528779128521276398678019,-0.6423493394403402206439846069955156500716,0.6423493394403402206439846069955156500716,-0.8015780907333099127942064895828598903056,0.8015780907333099127942064895828598903056,-0.9175983992229779652065478365007195123904,0.9175983992229779652065478365007195123904,-0.9841830547185881494728294488071096110649,0.9841830547185881494728294488071096110649],
  [-0.1080549487073436620662446502198347476119,0.1080549487073436620662446502198347476119,-0.3191123689278897604356718241684754668342,0.3191123689278897604356718241684754668342,-0.5152486363581540919652907185511886623088,0.5152486363581540919652907185511886623088,-0.6872929048116854701480198030193341375384,0.6872929048116854701480198030193341375384,-0.8272013150697649931897947426503949610397,0.8272013150697649931897947426503949610397,-0.9284348836635735173363911393778742644770,0.9284348836635735173363911393778742644770,-0.9862838086968123388415972667040528016760,0.9862838086968123388415972667040528016760],
  [0,-0.2011940939974345223006283033945962078128,0.2011940939974345223006283033945962078128,-0.3941513470775633698972073709810454683627,0.3941513470775633698972073709810454683627,-0.5709721726085388475372267372539106412383,0.5709721726085388475372267372539106412383,-0.7244177313601700474161860546139380096308,0.7244177313601700474161860546139380096308,-0.8482065834104272162006483207742168513662,0.8482065834104272162006483207742168513662,-0.9372733924007059043077589477102094712439,0.9372733924007059043077589477102094712439,-0.9879925180204854284895657185866125811469,0.9879925180204854284895657185866125811469],
  [-0.0950125098376374401853193354249580631303,0.0950125098376374401853193354249580631303,-0.2816035507792589132304605014604961064860,0.2816035507792589132304605014604961064860,-0.4580167776572273863424194429835775735400,0.4580167776572273863424194429835775735400,-0.6178762444026437484466717640487910189918,0.6178762444026437484466717640487910189918,-0.7554044083550030338951011948474422683538,0.7554044083550030338951011948474422683538,-0.8656312023878317438804678977123931323873,0.8656312023878317438804678977123931323873,-0.9445750230732325760779884155346083450911,0.9445750230732325760779884155346083450911,-0.9894009349916499325961541734503326274262,0.9894009349916499325961541734503326274262],
  [0,-0.1784841814958478558506774936540655574754,0.1784841814958478558506774936540655574754,-0.3512317634538763152971855170953460050405,0.3512317634538763152971855170953460050405,-0.5126905370864769678862465686295518745829,0.5126905370864769678862465686295518745829,-0.6576711592166907658503022166430023351478,0.6576711592166907658503022166430023351478,-0.7815140038968014069252300555204760502239,0.7815140038968014069252300555204760502239,-0.8802391537269859021229556944881556926234,0.8802391537269859021229556944881556926234,-0.9506755217687677612227169578958030214433,0.9506755217687677612227169578958030214433,-0.9905754753144173356754340199406652765077,0.9905754753144173356754340199406652765077],
  [-0.0847750130417353012422618529357838117333,0.0847750130417353012422618529357838117333,-0.2518862256915055095889728548779112301628,0.2518862256915055095889728548779112301628,-0.4117511614628426460359317938330516370789,0.4117511614628426460359317938330516370789,-0.5597708310739475346078715485253291369276,0.5597708310739475346078715485253291369276,-0.6916870430603532078748910812888483894522,0.6916870430603532078748910812888483894522,-0.8037049589725231156824174550145907971032,0.8037049589725231156824174550145907971032,-0.8926024664975557392060605911271455154078,0.8926024664975557392060605911271455154078,-0.9558239495713977551811958929297763099728,0.9558239495713977551811958929297763099728,-0.9915651684209309467300160047061507702525,0.9915651684209309467300160047061507702525],
  [0,-0.1603586456402253758680961157407435495048,0.1603586456402253758680961157407435495048,-0.3165640999636298319901173288498449178922,0.3165640999636298319901173288498449178922,-0.4645707413759609457172671481041023679762,0.4645707413759609457172671481041023679762,-0.6005453046616810234696381649462392798683,0.6005453046616810234696381649462392798683,-0.7209661773352293786170958608237816296571,0.7209661773352293786170958608237816296571,-0.8227146565371428249789224867127139017745,0.8227146565371428249789224867127139017745,-0.9031559036148179016426609285323124878093,0.9031559036148179016426609285323124878093,-0.9602081521348300308527788406876515266150,0.9602081521348300308527788406876515266150,-0.9924068438435844031890176702532604935893,0.9924068438435844031890176702532604935893],
  [-0.0765265211334973337546404093988382110047,0.0765265211334973337546404093988382110047,-0.2277858511416450780804961953685746247430,0.2277858511416450780804961953685746247430,-0.3737060887154195606725481770249272373957,0.3737060887154195606725481770249272373957,-0.5108670019508270980043640509552509984254,0.5108670019508270980043640509552509984254,-0.6360536807265150254528366962262859367433,0.6360536807265150254528366962262859367433,-0.7463319064601507926143050703556415903107,0.7463319064601507926143050703556415903107,-0.8391169718222188233945290617015206853296,0.8391169718222188233945290617015206853296,-0.9122344282513259058677524412032981130491,0.9122344282513259058677524412032981130491,-0.9639719272779137912676661311972772219120,0.9639719272779137912676661311972772219120,-0.9931285991850949247861223884713202782226,0.9931285991850949247861223884713202782226],
  [0,-0.1455618541608950909370309823386863301163,0.1455618541608950909370309823386863301163,-0.2880213168024010966007925160646003199090,0.2880213168024010966007925160646003199090,-0.4243421202074387835736688885437880520964,0.4243421202074387835736688885437880520964,-0.5516188358872198070590187967243132866220,0.5516188358872198070590187967243132866220,-0.6671388041974123193059666699903391625970,0.6671388041974123193059666699903391625970,-0.7684399634756779086158778513062280348209,0.7684399634756779086158778513062280348209,-0.8533633645833172836472506385875676702761,0.8533633645833172836472506385875676702761,-0.9200993341504008287901871337149688941591,0.9200993341504008287901871337149688941591,-0.9672268385663062943166222149076951614246,0.9672268385663062943166222149076951614246,-0.9937521706203895002602420359379409291933,0.9937521706203895002602420359379409291933],
  [-0.0697392733197222212138417961186280818222,0.0697392733197222212138417961186280818222,-0.2078604266882212854788465339195457342156,0.2078604266882212854788465339195457342156,-0.3419358208920842251581474204273796195591,0.3419358208920842251581474204273796195591,-0.4693558379867570264063307109664063460953,0.4693558379867570264063307109664063460953,-0.5876404035069115929588769276386473488776,0.5876404035069115929588769276386473488776,-0.6944872631866827800506898357622567712673,0.6944872631866827800506898357622567712673,-0.7878168059792081620042779554083515213881,0.7878168059792081620042779554083515213881,-0.8658125777203001365364256370193787290847,0.8658125777203001365364256370193787290847,-0.9269567721871740005206929392590531966353,0.9269567721871740005206929392590531966353,-0.9700604978354287271239509867652687108059,0.9700604978354287271239509867652687108059,-0.9942945854823992920730314211612989803930,0.9942945854823992920730314211612989803930],
  [0,-0.1332568242984661109317426822417661370104,0.1332568242984661109317426822417661370104,-0.2641356809703449305338695382833096029790,0.2641356809703449305338695382833096029790,-0.3903010380302908314214888728806054585780,0.3903010380302908314214888728806054585780,-0.5095014778460075496897930478668464305448,0.5095014778460075496897930478668464305448,-0.6196098757636461563850973116495956533871,0.6196098757636461563850973116495956533871,-0.7186613631319501944616244837486188483299,0.7186613631319501944616244837486188483299,-0.8048884016188398921511184069967785579414,0.8048884016188398921511184069967785579414,-0.8767523582704416673781568859341456716389,0.8767523582704416673781568859341456716389,-0.9329710868260161023491969890384229782357,0.9329710868260161023491969890384229782357,-0.9725424712181152319560240768207773751816,0.9725424712181152319560240768207773751816,-0.9947693349975521235239257154455743605736,0.9947693349975521235239257154455743605736],
  [-0.0640568928626056260850430826247450385909,0.0640568928626056260850430826247450385909,-0.1911188674736163091586398207570696318404,0.1911188674736163091586398207570696318404,-0.3150426796961633743867932913198102407864,0.3150426796961633743867932913198102407864,-0.4337935076260451384870842319133497124524,0.4337935076260451384870842319133497124524,-0.5454214713888395356583756172183723700107,0.5454214713888395356583756172183723700107,-0.6480936519369755692524957869107476266696,0.6480936519369755692524957869107476266696,-0.7401241915785543642438281030999784255232,0.7401241915785543642438281030999784255232,-0.8200019859739029219539498726697452080761,0.8200019859739029219539498726697452080761,-0.8864155270044010342131543419821967550873,0.8864155270044010342131543419821967550873,-0.9382745520027327585236490017087214496548,0.9382745520027327585236490017087214496548,-0.9747285559713094981983919930081690617411,0.9747285559713094981983919930081690617411,-0.9951872199970213601799974097007368118745,0.9951872199970213601799974097007368118745]
];

// Legendre-Gauss weights (wi values, defined by a function linked to in the Bezier primer article)
var cValues = [
  [],[],
  [1.0,1.0],
  [0.8888888888888888888888888888888888888888,0.5555555555555555555555555555555555555555,0.5555555555555555555555555555555555555555],
  [0.6521451548625461426269360507780005927646,0.6521451548625461426269360507780005927646,0.3478548451374538573730639492219994072353,0.3478548451374538573730639492219994072353],
  [0.5688888888888888888888888888888888888888,0.4786286704993664680412915148356381929122,0.4786286704993664680412915148356381929122,0.2369268850561890875142640407199173626432,0.2369268850561890875142640407199173626432],
  [0.3607615730481386075698335138377161116615,0.3607615730481386075698335138377161116615,0.4679139345726910473898703439895509948116,0.4679139345726910473898703439895509948116,0.1713244923791703450402961421727328935268,0.1713244923791703450402961421727328935268],
  [0.4179591836734693877551020408163265306122,0.3818300505051189449503697754889751338783,0.3818300505051189449503697754889751338783,0.2797053914892766679014677714237795824869,0.2797053914892766679014677714237795824869,0.1294849661688696932706114326790820183285,0.1294849661688696932706114326790820183285],
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];

// LUT for binomial coefficient arrays per curve order 'n'
var binomialCoefficients = [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1]];

// Look up what the binomial coefficient is for pair {n,k}
function binomials(n, k) {
  return binomialCoefficients[n][k];
};

/**
 * Compute the curve derivative (hodograph) at t.
 */
function getDerivative(derivative, t, vs) {
  // the derivative of any 't'-less function is zero.
  var n = vs.length - 1,
      _vs,
      value,
      k;
  if (n === 0) {
    return 0;
  }

// direct values? compute!
  if (derivative === 0) {
    value = 0;
    for (k = 0; k <= n; k++) {
      value += binomials(n, k) * Math.pow(1 - t, n - k) * Math.pow(t, k) * vs[k];
    }
    return value;
  } else {
    // Still some derivative? go down one order, then try
    // for the lower order curve's.
    _vs = new Array(n);
    for (k = 0; k < n; k++) {
      _vs[k] = n * (vs[k + 1] - vs[k]);
    }
    return getDerivative(derivative - 1, t, _vs);
  }
}

function B(xs, ys, t) {
  var xbase = getDerivative(1, t, xs);
  var ybase = getDerivative(1, t, ys);
  var combined = xbase * xbase + ybase * ybase;
  return Math.sqrt(combined);
}

function getArcLength(xs, ys, t, n) {
  var z, sum, i, correctedT;

if (xs.length >= tValues.length) {
    throw new Error('too high n bezier');
  }

if (t === undefined) {
    t = 1;
  }
  if (n === undefined) {
    n = 20;
  }

z = t / 2;
  sum = 0;
  for (i = 0; i < n; i++) {
    correctedT = z * tValues[n][i] + z;
    sum += cValues[n][i] * B(xs, ys, correctedT);
  }
  return z * sum;
}

updateCurveLengthText();

</script>
</body>
</html>