计算二次贝塞尔曲线的交点

// canvas vars
var canvas=document.getElementById("canvas");
var ctx=canvas.getContext("2d");
var cw=canvas.width;
var ch=canvas.height;

// linear interpolation utility
var lerp=function(a,b,x){ return(a+x*(b-a)); };

// qCurve & line defs
var p1={x:125,y:200};
var p2={x:250,y:225};
var p3={x:275,y:100};
var a1={x:30,y:125};
var a2={x:300,y:175};

// calc the intersections
var points=calcQLintersects(p1,p2,p3,a1,a2);

// plot the curve, line & solution(s)
var textPoints='Intersections: ';
ctx.beginPath();
ctx.moveTo(p1.x,p1.y);
ctx.quadraticCurveTo(p2.x,p2.y,p3.x,p3.y);
ctx.moveTo(a1.x,a1.y);
ctx.lineTo(a2.x,a2.y);
ctx.stroke();
ctx.beginPath();
for(var i=0;i<points.length;i++){
  var p=points[i];
  ctx.moveTo(p.x,p.y);
  ctx.arc(p.x,p.y,4,0,Math.PI*2);
  ctx.closePath();
  textPoints+=' ['+parseInt(p.x)+','+parseInt(p.y)+']';
}
ctx.font='14px verdana';
ctx.fillText(textPoints,10,20);
ctx.fillStyle='red';
ctx.fill();

///////////////////////////////////////////////////

function calcQLintersects(p1, p2, p3, a1, a2) {
  var intersections=[];

  // inverse line normal
  var normal={
    x: a1.y-a2.y,
    y: a2.x-a1.x,
  }

  // Q-coefficients
  var c2={
    x: p1.x + p2.x*-2 + p3.x,
    y: p1.y + p2.y*-2 + p3.y
  }

  var c1={
    x: p1.x*-2 + p2.x*2,
    y: p1.y*-2 + p2.y*2,
  }

  var c0={
    x: p1.x,
    y: p1.y
  }

  // Transform to line 
  var coefficient=a1.x*a2.y-a2.x*a1.y;
  var a=normal.x*c2.x + normal.y*c2.y;
  var b=(normal.x*c1.x + normal.y*c1.y)/a;
  var c=(normal.x*c0.x + normal.y*c0.y + coefficient)/a;

  // solve the roots
  var roots=[];
  d=b*b-4*c;
  if(d>0){
    var e=Math.sqrt(d);
    roots.push((-b+Math.sqrt(d))/2);
    roots.push((-b-Math.sqrt(d))/2);
  }else if(d==0){
    roots.push(-b/2);
  }

  // calc the solution points
  for(var i=0;i<roots.length;i++){
    var minX=Math.min(a1.x,a2.x);
    var minY=Math.min(a1.y,a2.y);
    var maxX=Math.max(a1.x,a2.x);
    var maxY=Math.max(a1.y,a2.y);
    var t = roots[i];
    if (t>=0 && t<=1) {
      // possible point -- pending bounds check
      var point={
        x:lerp(lerp(p1.x,p2.x,t),lerp(p2.x,p3.x,t),t),
        y:lerp(lerp(p1.y,p2.y,t),lerp(p2.y,p3.y,t),t)
      }
      var x=point.x;
      var y=point.y;
      // bounds checks
      if(a1.x==a2.x && y>=minY && y<=maxY){  
        // vertical line
        intersections.push(point);
      }else if(a1.y==a2.y && x>=minX && x<=maxX){
        // horizontal line
        intersections.push(point);
      }else if(x>=minX && y>=minY && x<=maxX && y<=maxY){
        // line passed bounds check
        intersections.push(point);
      }
    }
  }
  return intersections;
}

body{ background-color: ivory; padding:10px; }
#canvas{border:1px solid red;}

<h4>Calculate intersections of QBez-Curve and Line</h4>
<canvas id="canvas" width=350 height=350></canvas>