parabolic_arclength_solver.js

//Finds an unknown point distance L along a quadratic curve from a known point. 
//Alex Pilafian 2017 - [email protected] - github.com/sikanrong
//If you reuse this code please give me attribution, my dude! I worked hard on this.

//parabola defined by ax^2 + bx + c, a and b are passed in to constructor while c is omitted because it isn't relevant to our calculations.
//u is known point x-value
//L is known length to travel down the curve for our unknown point.
//v is the unknown point x-value, once we have v we can calculate the correspondiing unknown y just by pluging it 
//back into our parabola function

var sigFigs = 100000; //round to 5 significant figures
function ParabolicArcSolver(a, b, u, L){
  var du = (b + 2*a*u);
  var lu = Math.sqrt(1 + Math.pow(du, 2));
  
  var taylor = function(){
    return u + (L / lu)
  }
  
  var newton = function(v){
    var dv = (b + 2*a*v);
    var lv = Math.sqrt(1 + Math.pow(dv, 2));
    return v + ((lv*(4*a*L + du*lu - dv*lv + Math.asinh(du) - Math.asinh(dv)))/(2*a*(2 + 2 * Math.pow(dv, 2))))
  }
  
  //get a good first guess for newton from taylor
  var firstGuess = taylor();
  
  //Recursively run newton until it converges on an answer
  var doApproximation = function(v){
    var lastNewton = newton(v)
    if((v - lastNewton) <= 1/sigFigs){
      return (Math.floor(Math.round(v*sigFigs)) / sigFigs);
    }else{
      return doApproximation(lastNewton);
    }
  }
  
  return doApproximation(firstGuess);
}