Imitate hand-drawn (imperfect) circles using cubic Bézier segments

Imitate hand-drawn (imperfect) circles using cubic Bézier segments, based on a Bézier approximation to a perfect circle. TLDR: a good approximation to a quarter circle of unit radius is a cubic Bézier curve with control points P₀ = (0,1), P₁ = (c,1), P₂ = (1,c), P₃ = (1,0) where c = 0.551915024494. By varying the points in a sensible way, we can do a good impression of hand-drawn circles.

<!DOCTYPE html>
<html lang="en"> 
<head>
  <meta charset="utf-8"/>
  <style>
body {
  width: 600px;
  margin: auto;
}
input {
    width: 6em;
}
span.inputs {
    float: right;
}
svg {
    background: #eee;
}
svg.inset {
    float: left;
    margin-right: 20px;
}
#perturb0 {
    float: right;
    margin-left: 20px;
}
div.svg-group {
    text-align: center;
    margin: auto;
}
.svg-group svg {
    margin: 10px;
}
circle {
  stroke: none;
  fill: red;
}
path.pencil {
  stroke-width: 2px;
  stroke: #666;
  stroke-linecap: round;
  fill: none;
  vector-effect: non-scaling-stroke;
}
circle.reference {
  stroke: #ccc;
  stroke-width: 1px;
  vector-effect: non-scaling-stroke;
  fill: none;
}
  </style>
  <script src="https://d3js.org/d3.v3.min.js" charset="utf-8"></script>
  <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.0/jquery.min.js"></script>
  <script charset="utf-8">
function circlePath(δr_min,δr_max, θ0_min, θ0_max, δθ_min,δθ_max) {
    var c = 0.551915024494,
        β = Math.atan(c),
        d = Math.sqrt(c*c+1*1),
        r = 1,
        θ = (θ0_min + Math.random()*(θ0_max - θ0_min))*Math.PI/180,
        path = 'M';
    
    path += [r * Math.sin(θ), r * Math.cos(θ)];
    path += ' C' + [d * r * Math.sin(θ + β), d * r * Math.cos(θ + β)];
    
    for (var i=0; i<4; i++) {
        θ += Math.PI/2 * (1 + δθ_min + Math.random()*(δθ_max - δθ_min));
        r *= (1 + δr_min + Math.random()*(δr_max - δr_min));
        path += ' ' + (i?'S':'') + [d * r * Math.sin(θ - β), d * r * Math.cos(θ - β)];
        path += ' ' + [r * Math.sin(θ), r * Math.cos(θ)];
    }
    
    return path;
}
function circleXform(λ_min, λ_max, θ_min, θ_max) {
    var θ = (θ_min + Math.random()*(θ_max - θ_min));
    return 'rotate(' + θ + ') '
        + 'scale(1, ' + (λ_min + Math.random()*(λ_max - λ_min)) + ')'
        + 'rotate(' + (-θ) + ')';
}
  </script>
</head>
<body>
  <h1>Hand drawn circles</h1>
  <svg id='perturb0' width='100' height='100' viewBox="-1.5 -1.5 3 3"></svg>
  <p>
    I wanted to draw a month-at-a-glance calendar page with several days circled,
    but drawing perfect circles looked odd, so I looked for a way to imitiate hand-drawn
    (imperfect) circles.  
  </p>
  <p>
    For my application the circles don't need to be closed since I'm not filling them,
    but that would be an easy modification.
  </p>
  <svg class="inset" width='100' height='100' viewBox="-0.5 -0.5 2 2">
    <path class='pencil' d="M0,1 C0.552,1 1,0.552 1,0"/>
    <circle cx="0" cy="1" r="0.04"/>
    <circle cx="0.552" cy="1" r="0.04"/>
    <circle cx="1" cy=".552" r="0.04"/>
    <circle cx="1" cy="0" r="0.04"/>  
  </svg>
  <p>I started from this <a href="">nice approximation of a perfect circle</a> based on 
    cubic B&eacute;zier segments, where they show that a good approximation 
    to a quarter circle of unit radius is a cubic B&eacute;zier curve with control points
    <em>P<sub>0</sub></em> = (0, 1), <em>P<sub>1</sub></em> = (<em>c</em>, 1), 
    <em>P<sub>2</sub></em> = (1, <em>c</em>), <em>P<sub>3</sub></em> = (1, 0) where c = 0.551915024494.  
    For example we can use this SVG, with the control points shown in red: <code>&lt;path d="M0,1 C0.552,1 1,0.552 1,0"/></code>
  <p>
  <svg class="inset" width='100' height='100' viewBox="-1.5 -1.5 3 3">
    <path class='pencil' d="M0,1 C0.552,1 1,0.552 1,0 S0.552,-1 0,-1 S-1,-0.552 -1,0 S-0.552,1 0,1"/>
    <circle cx="0" cy="1" r="0.06"/>
    <circle cx="1" cy="0" r="0.06"/>
    <circle cx="0" cy="-1" r="0.06"/>
    <circle cx="-1" cy="0" r="0.06"/>
  </svg>
  <p>
    To draw a full circle, we just chain four of these together, using a path like this:
    <code>&lt;path d="M0,1 C0.552,1 1,0.552 1,0 S0.552,-1 0,-1 S-1,-0.552 -1,0 S-0.552,1 0,1"/></code>
  </p>
  <p>
    To make an approximate circle, it's more convenient to work in polar coordinates, 
    where we represent points using a radius and angle, (<em>r, &theta;</em>), 
    relative to the origin. 
    So the four control points for each arc, in polar coordinates, become
    <em>P<sub>0</sub></em> = (1, 0), <em>P<sub>1</sub></em> = (<em>d, &beta;</em>), 
    <em>P<sub>2</sub></em> = (<em>d, &pi;</em>/2 - <em>&beta;</em>), 
    <em>P<sub>3</sub></em> = (1, <em>&pi;</em>/2),
    where <em>&beta;</em> = tan<sup>-1</sup>(<em>c</em>) and 
    <em>d</em> = &radic; (<em>c</em><sup>2</sup> + 1).
  </p>
  <p>
    Now we just need to add a bit of randomness as we generate the path.
    I decided to do this in four ways:
    <ol>
      <li>Add a random delta to the current radius (starting from <em>r</em> = 1) 
      when we generate the start of each quarter 
      (along with its corresponding control points).
      If we bias the change to be slightly positive (or negative), the circle ends up
      looking more like a spiral.
      </li>
      <li>Start the path at a random angle.</li>
      <li>Add a random delta to the angle we rotate for each quarter, instead of
      exactly <em>&pi;</em>/2.  If we bias the change to be slightly positive
      (or negative) then the circle tends to overshoot (undershoot) the starting point.
      </li>
      <li>Squash the circle using a non-uniform scaling and random rotation.
      If we bias the scaling to be more or less than one we can make the circles tend to
      fit inside a fixed outline for example. 
    </ol>
    <div class="svg-group">
    <svg id='perturb1' width='100' height='100' viewBox="-1.5 -1.5 3 3"></svg>
    <svg id='perturb2' width='100' height='100' viewBox="-1.5 -1.5 3 3"></svg>
    <svg id='perturb3' width='100' height='100' viewBox="-1.5 -1.5 3 3"></svg>
    <svg id='perturb4' width='100' height='100' viewBox="-1.5 -1.5 3 3"></svg>
    </div>
    <script>
       d3.select('#perturb1')
        .append('path')
        .classed('pencil', true)
        .attr('d', circlePath(-0.1,0, 0,0, 0,0));
       d3.select('#perturb2')
        .append('path')
        .classed('pencil', true)
        .attr('d', circlePath(-0.1,0, 0,360, 0,0));
       d3.select('#perturb3')
        .append('path')
        .classed('pencil', true)
        .attr('d', circlePath(-0.1,0, 0,360, 0,0.2));
       d3.selectAll('#perturb4, #perturb0')
        .append('path')
        .classed('pencil', true)
        .attr('d', circlePath(-0.1,0, 0,360, 0,0.2))
        .attr('transform', circleXform(0.6, 0.8, 0, 360));
    </script>
    If we plan to draw lots of circles, we might want to choose the random variations 
    from a limited range so they all look like the same "handwriting".  Try experimenting:
    <p>Radius variation (0: none, &lt;0: shrink, &gt;0: grow)
      <span class='inputs'>
        <label for="drmin">min</label><input name='drmin' id='drmin' value='-0.1'>
        <label for="drmax">max</label><input name='drmax' id='drmax' value='0'>
      </span>
    </p>
    <p>Starting angle (0-360):
      <span class='inputs'>
        <label for="amin">min</label><input name='amin' id='amin' value='200'>
        <label for="amax">max</label><input name='amax' id='amax' value='240'>
      </span>
    </p>
    <p>Rotation variation (0: none, &lt;0: undershoot, &gt;0: overshoot)
      <span class='inputs'>
        <label for="damin">min</label><input name='damin' id='damin' value='0'>
        <label for="damax">max</label><input name='damax' id='damax' value='0.2'>
      </span>
    </p>
    <p>Squash factor (1: none, &lt;1: shrink, &gt;1: grow)
      <span class='inputs'>
        <label for="smin">min</label><input name='smin' id='smin' value='0.6'>
        <label for="smax">max</label><input name='smax' id='smax' value='0.8'>
      </span>
    </p>
    <p>Squash orientation (0-360)
      <span class='inputs'>
        <label for="samin">min</label><input name='samin' id='samin' value='30'>
        <label for="samax">max</label><input name='samax' id='samax' value='50'>
      </span>
    </p> 
    <div id="gallery"></div>
    <script>
d3.select('#gallery')
    .selectAll('div')
    .data(d3.range(5))
  .enter().append('div')
    .classed('svg-group',true)
    .selectAll('svg')
    .data(function(d) { return d3.range(5); })
  .enter().append('svg')
    .attr('width', 100)
    .attr('height', 100)
    .attr('viewBox', "-1.5 -1.5 3 3");

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

Modified http://d3js.org/d3.v3.min.js to a secure url

Modified http://ajax.googleapis.com/ajax/libs/jquery/1.11.0/jquery.min.js to a secure url

https://d3js.org/d3.v3.min.js

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