[加法回答:如何近似椭圆上的最近点]
如果你愿意牺牲完美的实用性...
这里是一种方法来计算一个椭圆点是“接近ish”到你的目标点。
的方法:
- 确定你的目标点在椭圆的象限
- 计算开始和结束该象限的弧度角。
- 沿椭圆象限计算点(“走椭圆”)。
- 对于每个计算出的椭圆点,计算到目标点的距离。
- 保存距目标最短距离的椭圆点。
缺点:
- 结果为近似值。
- 它比数学上完美的计算更不雅 - 使用暴力方法。
- (但一个有效的蛮力方法)。
优点:
- 近似的结果是相当不错的。
- 表现很不错。
- 计算简单得多。
- 计算(可能)比数学上完美的计算更快。
- (成本约20三角函数计算加上一些加法/减法)
- 如果需要更高的精度,你只需要改变1可变
- (更准确地花费更多的计算,当然)
性能注意:
- 你可以预先计算所有“走点”椭圆上的甚至更好的性能。
下面是这个方法的代码:
// 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});
}
这里是代码和一个小提琴: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>
原来的答案
这里的圆形和椭圆形如何相关的
对于水平对准椭圆:
(X X)/(一个一)+(Y Y)/(B B)== 1;
其中a
是水平顶点的长度,其中b
是垂直顶点的长度。
如何圆和椭圆涉及:
如果== B,椭圆是圆!
但是......!
计算从任意点到椭圆上某个点的最小距离涉及到多于的计算。
下面是计算一个链接(点击DistancePointEllipseEllipsoid.cpp):
http://www.geometrictools.com/SampleMathematics/DistancePointEllipseEllipsoid/DistancePointEllipseEllipsoid.html
一个椭圆有两个半径和旋转角度旁边的原点,所以你是什么意思“*射线意思是半径的一半*“? – Bergi
抱歉,ray是圆的半径。我不知道这转化为一个椭圆。 – Zero