Building a circle with quadratic curves in canvas

Question

I am trying to build a near-perfect circle with quadratic curves in canvas. I have this function for setting up points around a circle and connecting them with quadratic curves:

function calcPointsCircle(cx, cy, radius, dashLength) {
     var n = radius / dashLength,
         alpha = Math.PI * 2 / n,
         i = -1;
     while (i < n) {
         var theta = alpha * i,
             theta2 = alpha * (i + 1);
         points.push({
             x : (Math.cos(theta) * radius) + cx, 
             y : (Math.sin(theta) * radius) + cy, 
             ex : (Math.cos(theta2) * radius) + cx, 
             ey : (Math.sin(theta2) * radius) + cy,
             py : (Math.sin(theta) * radius) + cy
         });
         i+=2;
     }
}
for (i = 0; i < points.length; i++) {
    var p = points[i];
    ctx.strokeStyle = '#fff';
    ctx.quadraticCurveTo(p.x, p.py, p.x, p.y);
    ctx.stroke();
}

It works, but the lines are currently straight (which is obvious, since I am using the points x and y coordinates for the control point):

I am looking for a way to automatically calculate the poisitions for the control points based on the circle radius and the number of points... All help is more then welcome

Answer 1

Here's how to calculate the controls points of a set of quadratic curves which approximate a circle circumscribing a regular polygon.

Given:

A centerpoint, radius & sidecount.

For each side of the polygon, calculate:

3 points on a circumscribing circumference and then calculate the quadratic curve control point that causes the curve to pass through those 3 points:

• The 2 points of the polygon side are 2 of the 3 points

• Calculate the sweep angle between the 2 points of a side (var sweep)

• Bisect the sweep angle (sweep/2)

• Use trigonometry to calculate the point on the circumference midway between the 2 points of the side.

• Calculate the middle control point:

  // calc middle control point
  var cpX=2*x1-x0/2-x2/2;
  var cpY=2*y1-y0/2-y2/2;
  

Example code and a Demo:

// change sideCount to the # of poly sides desired
//
var sideCount=5;


var canvas=document.getElementById("canvas");
var ctx=canvas.getContext("2d");
ctx.lineWidth=2;
ctx.fillStyle=randomColor();

// save PI*2  
var PI2=Math.PI*2;

// functions to calc a point on circumference of circle
var xx=function(a){return(cx+radius*Math.cos(a));}
var yy=function(a){return(cy+radius*Math.sin(a));}

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

// define the regular polygon
var cx=150;
var cy=150;
var radius=100;

// calc qCurve controls points and put in sides[] array
var sides=[];
for(var i=0;i<sideCount;i++){
  sides.push(makeSide(i,sideCount));
}

// drawing and animating stuff
var percent=0;
var percentDirection=0.50;

$("#toShape").click(function(){
  percentDirection=-0.50;
})

$("#toCircle").click(function(){
  percentDirection=0.50;
})

animate();

// functions

function animate(){
  requestAnimationFrame(animate);
  drawSides(percent);
  percent+=percentDirection;
  if(percent>100){percent=100;}
  if(percent<0){percent=0;}
}


function drawSides(pct,color){
  ctx.clearRect(0,0,canvas.width,canvas.height);
  if(pct==100){
    ctx.beginPath();
    ctx.arc(cx,cy,radius,0,PI2);
    ctx.closePath();
    ctx.fill();
  }else{
    ctx.beginPath();
    ctx.moveTo(sides[0].x0,sides[0].y0);
    for(var i=0;i<sideCount;i++){
      var side=sides[i];
      var cpx=lerp(side.midX,side.cpX,pct/100);
      var cpy=lerp(side.midY,side.cpY,pct/100);        
      ctx.quadraticCurveTo(cpx,cpy,side.x2,side.y2);
    }
    ctx.fill();
  }
}

// given a side of a regular polygon,
// calc a qCurve that approximates a circle 
function makeSide(n,sideCount){

  // starting & ending angles vs centerpoint       
  var sweep=PI2/sideCount;
  var sAngle=sweep*(n-1);
  var eAngle=sweep*n;

  // given start & end points,
  // calc the point on circumference at middle of sweep angle
  var x0=xx(sAngle);
  var y0=yy(sAngle);
  var x1=xx((eAngle+sAngle)/2);
  var y1=yy((eAngle+sAngle)/2);
  var x2=xx(eAngle);
  var y2=yy(eAngle);

  // calc the control points to pass a qCurve 
  // through the 3 points
  var dx=x2-x1;
  var dy=y2-y1;
  var a=Math.atan2(dy,dx);
  var midX=lerp(x0,x2,0.50);
  var midY=lerp(y0,y2,0.50);

  // calc middle control point            
  var cpX=2*x1-x0/2-x2/2;
  var cpY=2*y1-y0/2-y2/2;

  return({
    x0:x0, y0:y0,
    x2:x2, y2:y2,
    midX:midX, midY:midY,
    cpX:cpX, cpY:cpY,
    color:randomColor()
  });
}

function randomColor(){ 
  return('#'+Math.floor(Math.random()*16777215).toString(16));
}

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

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.9.1/jquery.min.js"></script>
<button id="toShape">Animate to Shape</button>
<button id="toCircle">Animate to Circle</button><br>
<canvas id="canvas" width=300 height=300></canvas>