计算两条线的交点

Question

我有动态生成动画的线条，我想检测线条何时撞到另一条线条。我正在尝试实现一些基本的线性代数来获得线的方程，然后求解 x,y，但结果不稳定。在这一点上，我只用两条线进行测试，这意味着我应该得到一个交点，但我得到了两个。我只是想确保我的数学没问题，我应该在别处寻找问题。

function collision(boid1, boid2) {
  var x1 = boid1.initialX, y1 = boid1.initialY, x2 = boid1.x, y2 = boid1.y, x3 = boid2.initialX, y3 = boid2.initialY, x4 = boid2.x, y4 = boid2.y;
  slope1 = (y1 - y2)/(x1 - x2);
  slope2 = (y3 - y4)/(x3- x4);

  if(slope1 != slope2){
    var b1 = getB(slope1,x1,y1);
    var b2 = getB(slope2,x3,y3);

    if(slope2 >= 0){
      u = slope1 - slope2;
    }else{
      u = slope1 + slope2;
    }

    if(b1 >= 0){
      z = b2 - b1;
    }else{
      z = b2 + b1;
    }
    pointX = z / u;
    pointY = (slope1*pointX)+b1;
    pointYOther = (slope2*pointX)+b2;
    console.log("pointx:"+pointX+" pointy:"+pointY+" othery:"+pointYOther);
    context.beginPath();
    context.arc(pointX, pointY, 2, 0, 2 * Math.PI, false);
    context.fillStyle = 'green';
    context.fill();
    context.lineWidth = 1;
    context.strokeStyle = '#003300';
    context.stroke();
  }
  return false;
}

function getB(slope,x,y){
  var y = y, x = x, m = slope;
  a = m*x;
  if(a>=0){
    b = y - a;
  }else{
    b = y + a;
  }
  return b;
}

问题是我得到了两个不同的交点值。应该只有一个，这让我相信我的计算是错误的。是的，x2,y2,x4,y4 都在移动，但它们有一个固定的角​​度，一致的坡度证实了这一点。

Answer 1

我找到了great solution by Paul Bourke。在这里，用 JavaScript 实现：

function line_intersect(x1, y1, x2, y2, x3, y3, x4, y4)
{
    var ua, ub, denom = (y4 - y3)*(x2 - x1) - (x4 - x3)*(y2 - y1);
    if (denom == 0) {
        return null;
    }
    ua = ((x4 - x3)*(y1 - y3) - (y4 - y3)*(x1 - x3))/denom;
    ub = ((x2 - x1)*(y1 - y3) - (y2 - y1)*(x1 - x3))/denom;
    return {
        x: x1 + ua * (x2 - x1),
        y: y1 + ua * (y2 - y1),
        seg1: ua >= 0 && ua <= 1,
        seg2: ub >= 0 && ub <= 1
    };
}

Answer 2

对于线段-线段交点，使用Paul Bourke's solution:

// line intercept math by Paul Bourke http://paulbourke.net/geometry/pointlineplane/
// Determine the intersection point of two line segments
// Return FALSE if the lines don't intersect
function intersect(x1, y1, x2, y2, x3, y3, x4, y4) {

  // Check if none of the lines are of length 0
    if ((x1 === x2 && y1 === y2) || (x3 === x4 && y3 === y4)) {
        return false
    }

    denominator = ((y4 - y3) * (x2 - x1) - (x4 - x3) * (y2 - y1))

  // Lines are parallel
    if (denominator === 0) {
        return false
    }

    let ua = ((x4 - x3) * (y1 - y3) - (y4 - y3) * (x1 - x3)) / denominator
    let ub = ((x2 - x1) * (y1 - y3) - (y2 - y1) * (x1 - x3)) / denominator

  // is the intersection along the segments
    if (ua < 0 || ua > 1 || ub < 0 || ub > 1) {
        return false
    }

  // Return a object with the x and y coordinates of the intersection
    let x = x1 + ua * (x2 - x1)
    let y = y1 + ua * (y2 - y1)

    return {x, y}
}

对于无限的线交点，使用Justin C. Round's algorithm:

function checkLineIntersection(line1StartX, line1StartY, line1EndX, line1EndY, line2StartX, line2StartY, line2EndX, line2EndY) {
    // if the lines intersect, the result contains the x and y of the intersection (treating the lines as infinite) and booleans for whether line segment 1 or line segment 2 contain the point
    var denominator, a, b, numerator1, numerator2, result = {
        x: null,
        y: null,
        onLine1: false,
        onLine2: false
    };
    denominator = ((line2EndY - line2StartY) * (line1EndX - line1StartX)) - ((line2EndX - line2StartX) * (line1EndY - line1StartY));
    if (denominator == 0) {
        return result;
    }
    a = line1StartY - line2StartY;
    b = line1StartX - line2StartX;
    numerator1 = ((line2EndX - line2StartX) * a) - ((line2EndY - line2StartY) * b);
    numerator2 = ((line1EndX - line1StartX) * a) - ((line1EndY - line1StartY) * b);
    a = numerator1 / denominator;
    b = numerator2 / denominator;

    // if we cast these lines infinitely in both directions, they intersect here:
    result.x = line1StartX + (a * (line1EndX - line1StartX));
    result.y = line1StartY + (a * (line1EndY - line1StartY));

    // if line1 is a segment and line2 is infinite, they intersect if:
    if (a > 0 && a < 1) {
        result.onLine1 = true;
    }
    // if line2 is a segment and line1 is infinite, they intersect if:
    if (b > 0 && b < 1) {
        result.onLine2 = true;
    }
    // if line1 and line2 are segments, they intersect if both of the above are true
    return result;
};

Answer 3

将 'found-x' 插回方程式之一时，您无需在加/减 y 相交之间交替：

(function () {
    window.linear = {
        slope: function (x1, y1, x2, y2) {
            if (x1 == x2) return false;
            return (y1 - y2) / (x1 - x2);
        },
        yInt: function (x1, y1, x2, y2) {
            if (x1 === x2) return y1 === 0 ? 0 : false;
            if (y1 === y2) return y1;
            return y1 - this.slope(x1, y1, x2, y2) * x1 ;
        },
        getXInt: function (x1, y1, x2, y2) {
            var slope;
            if (y1 === y2) return x1 == 0 ? 0 : false;
            if (x1 === x2) return x1;
            return (-1 * ((slope = this.slope(x1, y1, x2, y2)) * x1 - y1)) / slope;
        },
        getIntersection: function (x11, y11, x12, y12, x21, y21, x22, y22) {
            var slope1, slope2, yint1, yint2, intx, inty;
            if (x11 == x21 && y11 == y21) return [x11, y11];
            if (x12 == x22 && y12 == y22) return [x12, y22];

            slope1 = this.slope(x11, y11, x12, y12);
            slope2 = this.slope(x21, y21, x22, y22);
            if (slope1 === slope2) return false;

            yint1 = this.yInt(x11, y11, x12, y12);
            yint2 = this.yInt(x21, y21, x22, y22);
            if (yint1 === yint2) return yint1 === false ? false : [0, yint1];

            if (slope1 === false) return [y21, slope2 * y21 + yint2];
            if (slope2 === false) return [y11, slope1 * y11 + yint1];
            intx = (slope1 * x11 + yint1 - yint2)/ slope2;
            return [intx, slope1 * intx + yint1];
        }
    }
}());

Answer 4

你可以这样做；

function lineIntersect(a,b){
  a.m = (a[0].y-a[1].y)/(a[0].x-a[1].x);  // slope of line 1
  b.m = (b[0].y-b[1].y)/(b[0].x-b[1].x);  // slope of line 2
  return a.m - b.m < Number.EPSILON ? undefined
                                    : { x: (a.m * a[0].x - b.m*b[0].x + b[0].y - a[0].y) / (a.m - b.m),
                                        y: (a.m*b.m*(b[0].x-a[0].x) + b.m*a[0].y - a.m*b[0].y) / (b.m - a.m)};
}

var line1 = [{x:3, y:3},{x:17, y:8}],
    line2 = [{x:7, y:10},{x:11, y:2}];
console.log(lineIntersect(line1, line2));

Answer 5

有一个 npm 模块可以做到这一点：line-intersect。

使用安装它

npm install --save line-intersect

ES6 用法：

import { checkIntersection } from "line-intersect";

const result = lineIntersect.checkIntersection(
  line1.start.x, line1.start.y, line1.end.x, line1.end.y,
  line2.start.x, line2.start.y, line2.end.x, line2.end.y
);

result.type  // any of "none", "parallel", "colinear", "intersecting"
result.point // only exists when result.type == 'intersecting'

如果您使用的是打字稿，这里是打字：

declare module "line-intersect" {
  export function checkIntersection(
    x1: number, y1: number,
    x2: number, y2: number,
    x3: number, y3: number,
    x4: number, y4: number): {
        type: string,
        point: {x:number, y:number}
    }; 
}

把它放在一个文件中，如果在tsconfig.json的"files"部分中引用。