[ 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.
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:
(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
