Converting a Circle to Ellipse so it calculates the Distance of a Point from an Ellipse Border

StackOverflow https://stackoverflow.com/questions/18362533

Question

I'm looking for the equation to convert a circle to an ellipse so that I can find the shortest distance from a point to an ellipse border. I have found the equation for the distance between a circle and a point but cant figure out how to convert it to work with a ellipse.

px and py are the points and x and y are the circle origin and ray is the radius

closestCirclePoint: function(px, py, x, y, ray) {
    var tg = (x += ray, y += ray, 0);
    return function(x, y, x0, y0) {
        return Math.sqrt((x -= x0) * x + (y -= y0) * y);
    }(px, py, x, y) > ray
      ? {x: Math.cos(tg = Math.atan2(py - y, px - x)) * ray + x,
         y: Math.sin(tg) * ray + y}
      : {x: px, y: py};
}
Was it helpful?

Solution

[ Addition to answer: How to approximate the nearest point on the ellipse]

If you are willing to sacrifice perfection for practicality…

Here is a way to calculate an ellipse point that is “near-ish” to your targeted point.

enter image description here

The method:

  • Determine which quadrant of the ellipse your target point is in.
  • Calculate the beginning and ending radian angles of that quadrant.
  • Calculate points along that ellipse quadrant (“walk the ellipse”).
  • For each calculated ellipse point, calc the distance to the target point.
  • Save the ellipse point with the shortest distance to the target.

Cons:

  • The result is approximate.
  • It's less elegant than the mathematically perfect calculation—uses a brute force method.
  • (but an efficient brute force method).

Pros:

  • The approximated result is pretty good.
  • Performance is pretty good.
  • The calculations are much simpler.
  • The calculations are (probably) faster than the mathematically perfect calculation.
  • (costs about 20 trig calculations plus some addition/subtraction)
  • If you need greater accuracy, you just change 1 variable
  • (greater accuracy costs more calculations, of course)

Performance note:

  • You could pre-calculate all the "walking points" on the ellipse for even better performance.

Here’s the code for this method:

    // calc a point on the ellipse that is "near-ish" the target point
    // uses "brute force"
    function getEllipsePt(targetPtX,targetPtY){

        // calculate which ellipse quadrant the targetPt is in
        var q;
        if(targetPtX>cx){
            q=(targetPtY>cy)?0:3;
        }else{
            q=(targetPtY>cy)?1:2;
        }

        // calc beginning and ending radian angles to check
        var r1=q*halfPI;
        var r2=(q+1)*halfPI;
        var dr=halfPI/steps;
        var minLengthSquared=200000000;
        var minX,minY;

        // walk the ellipse quadrant and find a near-point
        for(var r=r1;r<r2;r+=dr){

            // get a point on the ellipse at radian angle == r
            var ellipseX=cx+radiusX*Math.cos(r);
            var ellipseY=cy+radiusY*Math.sin(r);

            // calc distance from ellipsePt to targetPt
            var dx=targetPtX-ellipseX;
            var dy=targetPtY-ellipseY;
            var lengthSquared=dx*dx+dy*dy;

            // if new length is shortest, save this ellipse point
            if(lengthSquared<minLengthSquared){
                minX=ellipseX;
                minY=ellipseY;
                minLengthSquared=lengthSquared;
            }
        }

        return({x:minX,y:minY});
    }

Here is code and a Fiddle: http://jsfiddle.net/m1erickson/UDBkV/

<!doctype html>
<html>
<head>
<link rel="stylesheet" type="text/css" media="all" href="css/reset.css" /> <!-- reset css -->
<script type="text/javascript" src="http://code.jquery.com/jquery.min.js"></script>

<style>
    body{ background-color: ivory; padding:20px; }
    #wrapper{
        position:relative;
        width:300px;
        height:300px;
    }
    #canvas{
        position:absolute; top:0px; left:0px;
        border:1px solid green;
        width:100%;
        height:100%;
    }
    #canvas2{
        position:absolute; top:0px; left:0px;
        border:1px solid red;
        width:100%;
        height:100%;
    }
</style>

<script>
$(function(){

    // get canvas references
    var canvas=document.getElementById("canvas");
    var ctx=canvas.getContext("2d");
    var canvas2=document.getElementById("canvas2");
    var ctx2=canvas2.getContext("2d");

    // calc canvas position on page
    var canvasOffset=$("#canvas").offset();
    var offsetX=canvasOffset.left;
    var offsetY=canvasOffset.top;


    // define the ellipse
    var cx=150;
    var cy=150;
    var radiusX=50;
    var radiusY=25;
    var halfPI=Math.PI/2;
    var steps=8; // larger == greater accuracy


    // get mouse position
    // calc a point on the ellipse that is "near-ish"
    // display a line between the mouse and that ellipse point
    function handleMouseMove(e){
      mouseX=parseInt(e.clientX-offsetX);
      mouseY=parseInt(e.clientY-offsetY);

      // Put your mousemove stuff here
      var pt=getEllipsePt(mouseX,mouseY);

      // testing: draw results
      drawResults(mouseX,mouseY,pt.x,pt.y);
    }


    // calc a point on the ellipse that is "near-ish" the target point
    // uses "brute force"
    function getEllipsePt(targetPtX,targetPtY){

        // calculate which ellipse quadrant the targetPt is in
        var q;
        if(targetPtX>cx){
            q=(targetPtY>cy)?0:3;
        }else{
            q=(targetPtY>cy)?1:2;
        }

        // calc beginning and ending radian angles to check
        var r1=q*halfPI;
        var r2=(q+1)*halfPI;
        var dr=halfPI/steps;
        var minLengthSquared=200000000;
        var minX,minY;

        // walk the ellipse quadrant and find a near-point
        for(var r=r1;r<r2;r+=dr){

            // get a point on the ellipse at radian angle == r
            var ellipseX=cx+radiusX*Math.cos(r);
            var ellipseY=cy+radiusY*Math.sin(r);

            // calc distance from ellipsePt to targetPt
            var dx=targetPtX-ellipseX;
            var dy=targetPtY-ellipseY;
            var lengthSquared=dx*dx+dy*dy;

            // if new length is shortest, save this ellipse point
            if(lengthSquared<minLengthSquared){
                minX=ellipseX;
                minY=ellipseY;
                minLengthSquared=lengthSquared;
            }
        }

        return({x:minX,y:minY});
    }

    // listen for mousemoves
    $("#canvas").mousemove(function(e){handleMouseMove(e);});



    // testing: draw the ellipse on the background canvas
    function drawEllipse(){
        ctx2.beginPath()
        ctx2.moveTo(cx+radiusX,cy)
        for(var r=0;r<2*Math.PI;r+=2*Math.PI/60){
            var ellipseX=cx+radiusX*Math.cos(r);
            var ellipseY=cy+radiusY*Math.sin(r);
            ctx2.lineTo(ellipseX,ellipseY)
        }
        ctx2.closePath();
        ctx2.lineWidth=5;
        ctx2.stroke();
    }

    // testing: draw line from mouse to ellipse
    function drawResults(mouseX,mouseY,ellipseX,ellipseY){
        ctx.clearRect(0,0,canvas.width,canvas.height);
        ctx.beginPath();
        ctx.moveTo(mouseX,mouseY);
        ctx.lineTo(ellipseX,ellipseY);
        ctx.lineWidth=1;
        ctx.strokeStyle="red";
        ctx.stroke();
    }


}); // end $(function(){});
</script>

</head>

<body>
    <div id="wrapper">
        <canvas id="canvas2" width=300 height=300></canvas>
        <canvas id="canvas" width=300 height=300></canvas>
    </div>
</body>
</html>

Original Answer

Here's how circles and ellipses are related

For a horizontally aligned ellipse:

enter image description here

(xx) / (aa) + (yy) / (bb) == 1;

where a is the length to the horizontal vertex and where b is the length to the vertical vertex.

How circles and ellipses relate:

If a==b, the ellipse is a circle !

However...!

Calculating the minimal distance from any point to a point on an ellipse involves much more calculation than with a circle.

Here's a link to the calculation (click on DistancePointEllipseEllipsoid.cpp):

http://www.geometrictools.com/SampleMathematics/DistancePointEllipseEllipsoid/DistancePointEllipseEllipsoid.html

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