' VB.NET - Code

Function CheckLineSegmentCircleIntersection(x1 As Double, y1 As Double, x2 As Double, y2 As Double, xc As Double, yc As Double, r As Double) As Boolean
    Static xd As Double = 0.0F
    Static yd As Double = 0.0F
    Static t As Double = 0.0F
    Static d As Double = 0.0F
    Static dx_2_1 As Double = 0.0F
    Static dy_2_1 As Double = 0.0F

    dx_2_1 = x2 - x1
    dy_2_1 = y2 - y1

    t = ((yc - y1) * dy_2_1 + (xc - x1) * dx_2_1) / (dy_2_1 * dy_2_1 + dx_2_1 * dx_2_1)

    If 0 <= t And t <= 1 Then
        xd = x1 + t * dx_2_1
        yd = y1 + t * dy_2_1

        d = Math.Sqrt((xd - xc) * (xd - xc) + (yd - yc) * (yd - yc))
        Return d <= r
    Else
        d = Math.Sqrt((xc - x1) * (xc - x1) + (yc - y1) * (yc - y1))
        If d <= r Then
            Return True
        Else
            d = Math.Sqrt((xc - x2) * (xc - x2) + (yc - y2) * (yc - y2))
            If d <= r Then
                Return True
            Else
                Return False
            End If
        End If
    End If
End Function

似乎没人考虑投影，我是不是完全跑题了?

将向量AC投影到AB上，投影的向量AD就得到了新的点D。 如果D和C之间的距离小于(或等于)R，我们有一个交点。

是这样的:

社区编辑:

对于稍后无意中看到这篇文章并想知道如何实现这样一个算法的人来说，这里是一个使用常见向量操作函数用JavaScript编写的通用实现。

/**
 * Returns the distance from line segment AB to point C
 */
function distanceSegmentToPoint(A, B, C) {
    // Compute vectors AC and AB
    const AC = sub(C, A);
    const AB = sub(B, A);

    // Get point D by taking the projection of AC onto AB then adding the offset of A
    const D = add(proj(AC, AB), A);

    const AD = sub(D, A);
    // D might not be on AB so calculate k of D down AB (aka solve AD = k * AB)
    // We can use either component, but choose larger value to reduce the chance of dividing by zero
    const k = Math.abs(AB.x) > Math.abs(AB.y) ? AD.x / AB.x : AD.y / AB.y;

    // Check if D is off either end of the line segment
    if (k <= 0.0) {
        return Math.sqrt(hypot2(C, A));
    } else if (k >= 1.0) {
        return Math.sqrt(hypot2(C, B));
    }

    return Math.sqrt(hypot2(C, D));
}

对于这个实现，我使用了两个常见的矢量操作函数，无论您在什么环境中工作，都可能已经提供了这些函数。但是，如果您还没有这些可用的功能，下面介绍如何实现它们。

// Define some common functions for working with vectors
const add = (a, b) => ({x: a.x + b.x, y: a.y + b.y});
const sub = (a, b) => ({x: a.x - b.x, y: a.y - b.y});
const dot = (a, b) => a.x * b.x + a.y * b.y;
const hypot2 = (a, b) => dot(sub(a, b), sub(a, b));

// Function for projecting some vector a onto b
function proj(a, b) {
    const k = dot(a, b) / dot(b, b);
    return {x: k * b.x, y: k * b.y};
}

另一个在c#(部分圆类)。 经过测试，工作就像一个魅力。

public class Circle : IEquatable<Circle>
{
    // ******************************************************************
    // The center of a circle
    private Point _center;
    // The radius of a circle
    private double _radius;

   // ******************************************************************
    /// <summary>
    /// Find all intersections (0, 1, 2) of the circle with a line defined by its 2 points.
    /// Using: http://math.stackexchange.com/questions/228841/how-do-i-calculate-the-intersections-of-a-straight-line-and-a-circle
    /// Note: p is the Center.X and q is Center.Y
    /// </summary>
    /// <param name="linePoint1"></param>
    /// <param name="linePoint2"></param>
    /// <returns></returns>
    public List<Point> GetIntersections(Point linePoint1, Point linePoint2)
    {
        List<Point> intersections = new List<Point>();

        double dx = linePoint2.X - linePoint1.X;

        if (dx.AboutEquals(0)) // Straight vertical line
        {
            if (linePoint1.X.AboutEquals(Center.X - Radius) || linePoint1.X.AboutEquals(Center.X + Radius))
            {
                Point pt = new Point(linePoint1.X, Center.Y);
                intersections.Add(pt);
            }
            else if (linePoint1.X > Center.X - Radius && linePoint1.X < Center.X + Radius)
            {
                double x = linePoint1.X - Center.X;

                Point pt = new Point(linePoint1.X, Center.Y + Math.Sqrt(Radius * Radius - (x * x)));
                intersections.Add(pt);

                pt = new Point(linePoint1.X, Center.Y - Math.Sqrt(Radius * Radius - (x * x)));
                intersections.Add(pt);
            }

            return intersections;
        }

        // Line function (y = mx + b)
        double dy = linePoint2.Y - linePoint1.Y;
        double m = dy / dx;
        double b = linePoint1.Y - m * linePoint1.X;

        double A = m * m + 1;
        double B = 2 * (m * b - m * _center.Y - Center.X);
        double C = Center.X * Center.X + Center.Y * Center.Y - Radius * Radius - 2 * b * Center.Y + b * b;

        double discriminant = B * B - 4 * A * C;

        if (discriminant < 0)
        {
            return intersections; // there is no intersections
        }

        if (discriminant.AboutEquals(0)) // Tangeante (touch on 1 point only)
        {
            double x = -B / (2 * A);
            double y = m * x + b;

            intersections.Add(new Point(x, y));
        }
        else // Secant (touch on 2 points)
        {
            double x = (-B + Math.Sqrt(discriminant)) / (2 * A);
            double y = m * x + b;
            intersections.Add(new Point(x, y));

            x = (-B - Math.Sqrt(discriminant)) / (2 * A);
            y = m * x + b;
            intersections.Add(new Point(x, y));
        }

        return intersections;
    }

    // ******************************************************************
    // Get the center
    [XmlElement("Center")]
    public Point Center
    {
        get { return _center; }
        set
        {
            _center = value;
        }
    }

    // ******************************************************************
    // Get the radius
    [XmlElement]
    public double Radius
    {
        get { return _radius; }
        set { _radius = value; }
    }

    //// ******************************************************************
    //[XmlArrayItemAttribute("DoublePoint")]
    //public List<Point> Coordinates
    //{
    //    get { return _coordinates; }
    //}

    // ******************************************************************
    // Construct a circle without any specification
    public Circle()
    {
        _center.X = 0;
        _center.Y = 0;
        _radius = 0;
    }

    // ******************************************************************
    // Construct a circle without any specification
    public Circle(double radius)
    {
        _center.X = 0;
        _center.Y = 0;
        _radius = radius;
    }

    // ******************************************************************
    // Construct a circle with the specified circle
    public Circle(Circle circle)
    {
        _center = circle._center;
        _radius = circle._radius;
    }

    // ******************************************************************
    // Construct a circle with the specified center and radius
    public Circle(Point center, double radius)
    {
        _center = center;
        _radius = radius;
    }

    // ******************************************************************
    // Construct a circle based on one point
    public Circle(Point center)
    {
        _center = center;
        _radius = 0;
    }

    // ******************************************************************
    // Construct a circle based on two points
    public Circle(Point p1, Point p2)
    {
        Circle2Points(p1, p2);
    }

要求:

using System;

namespace Mathematic
{
    public static class DoubleExtension
    {
        // ******************************************************************
        // Base on Hans Passant Answer on:
        // http://stackoverflow.com/questions/2411392/double-epsilon-for-equality-greater-than-less-than-less-than-or-equal-to-gre

        /// <summary>
        /// Compare two double taking in account the double precision potential error.
        /// Take care: truncation errors accumulate on calculation. More you do, more you should increase the epsilon.
        public static bool AboutEquals(this double value1, double value2)
        {
            if (double.IsPositiveInfinity(value1))
                return double.IsPositiveInfinity(value2);

            if (double.IsNegativeInfinity(value1))
                return double.IsNegativeInfinity(value2);

            if (double.IsNaN(value1))
                return double.IsNaN(value2);

            double epsilon = Math.Max(Math.Abs(value1), Math.Abs(value2)) * 1E-15;
            return Math.Abs(value1 - value2) <= epsilon;
        }

        // ******************************************************************
        // Base on Hans Passant Answer on:
        // http://stackoverflow.com/questions/2411392/double-epsilon-for-equality-greater-than-less-than-less-than-or-equal-to-gre

        /// <summary>
        /// Compare two double taking in account the double precision potential error.
        /// Take care: truncation errors accumulate on calculation. More you do, more you should increase the epsilon.
        /// You get really better performance when you can determine the contextual epsilon first.
        /// </summary>
        /// <param name="value1"></param>
        /// <param name="value2"></param>
        /// <param name="precalculatedContextualEpsilon"></param>
        /// <returns></returns>
        public static bool AboutEquals(this double value1, double value2, double precalculatedContextualEpsilon)
        {
            if (double.IsPositiveInfinity(value1))
                return double.IsPositiveInfinity(value2);

            if (double.IsNegativeInfinity(value1))
                return double.IsNegativeInfinity(value2);

            if (double.IsNaN(value1))
                return double.IsNaN(value2);

            return Math.Abs(value1 - value2) <= precalculatedContextualEpsilon;
        }

        // ******************************************************************
        public static double GetContextualEpsilon(this double biggestPossibleContextualValue)
        {
            return biggestPossibleContextualValue * 1E-15;
        }

        // ******************************************************************
        /// <summary>
        /// Mathlab equivalent
        /// </summary>
        /// <param name="dividend"></param>
        /// <param name="divisor"></param>
        /// <returns></returns>
        public static double Mod(this double dividend, double divisor)
        {
            return dividend - System.Math.Floor(dividend / divisor) * divisor;
        }

        // ******************************************************************
    }
}

我发现这个解决方案似乎比其他一些解决方案更容易遵循。

采取:

p1 and p2 as the points for the line, and
c as the center point for the circle and r for the radius

我可以用斜截式来解直线方程。但是，我不想处理以c为点的复杂方程，所以我只是平移了坐标系使圆在(0,0)处

p3 = p1 - c
p4 = p2 - c

顺便说一下，当我相互减分的时候，我是在减去x再减去y，然后把它们放到一个新的点里，以防有人不知道。

不管怎样，我现在解出p3和p4的直线方程

m = (p4_y - p3_y) / (p4_x - p3) (the underscore is an attempt at subscript)
y = mx + b
y - mx = b (just put in a point for x and y, and insert the m we found)

好的。现在我需要让这两个方程相等。首先我需要解圆的x方程

x^2 + y^2 = r^2
y^2 = r^2 - x^2
y = sqrt(r^2 - x^2)

然后我让它们相等:

mx + b = sqrt(r^2 - x^2)

求二次方程(0 = ax^2 + bx + c)

(mx + b)^2 = r^2 - x^2
(mx)^2 + 2mbx + b^2 = r^2 - x^2
0 = m^2 * x^2 + x^2 + 2mbx + b^2 - r^2
0 = (m^2 + 1) * x^2 + 2mbx + b^2 - r^2

现在我有了a b c。

a = m^2 + 1
b = 2mb
c = b^2 - r^2

我把这个代入二次公式

(-b ± sqrt(b^2 - 4ac)) / 2a

用值代入，然后尽可能简化:

(-2mb ± sqrt(b^2 - 4ac)) / 2a
(-2mb ± sqrt((-2mb)^2 - 4(m^2 + 1)(b^2 - r^2))) / 2(m^2 + 1)
(-2mb ± sqrt(4m^2 * b^2 - 4(m^2 * b^2 - m^2 * r^2 + b^2 - r^2))) / 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * b^2 - (m^2 * b^2 - m^2 * r^2 + b^2 - r^2))))/ 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * b^2 - m^2 * b^2 + m^2 * r^2 - b^2 + r^2)))/ 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * r^2 - b^2 + r^2)))/ 2m^2 + 2
(-2mb ± sqrt(4) * sqrt(m^2 * r^2 - b^2 + r^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(m^2 * r^2 - b^2 + r^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(m^2 * r^2 + r^2 - b^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2

这几乎是化简的极限了。最后，分离出带有±的方程:

(-2mb + 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2 or     
(-2mb - 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2 

然后简单地将这两个方程的结果代入mx + b中的x。为了清晰起见，我写了一些JavaScript代码来演示如何使用这个:

function interceptOnCircle(p1,p2,c,r){
    //p1 is the first line point
    //p2 is the second line point
    //c is the circle's center
    //r is the circle's radius

    var p3 = {x:p1.x - c.x, y:p1.y - c.y} //shifted line points
    var p4 = {x:p2.x - c.x, y:p2.y - c.y}

    var m = (p4.y - p3.y) / (p4.x - p3.x); //slope of the line
    var b = p3.y - m * p3.x; //y-intercept of line

    var underRadical = Math.pow((Math.pow(r,2)*(Math.pow(m,2)+1)),2)-Math.pow(b,2)); //the value under the square root sign 

    if (underRadical < 0){
    //line completely missed
        return false;
    } else {
        var t1 = (-2*m*b+2*Math.sqrt(underRadical))/(2 * Math.pow(m,2) + 2); //one of the intercept x's
        var t2 = (-2*m*b-2*Math.sqrt(underRadical))/(2 * Math.pow(m,2) + 2); //other intercept's x
        var i1 = {x:t1,y:m*t1+b} //intercept point 1
        var i2 = {x:t2,y:m*t2+b} //intercept point 2
        return [i1,i2];
    }
}

我希望这能有所帮助!

附注:如果任何人发现任何错误或有任何建议，请评论。我是新手，欢迎大家的帮助/建议。

只是这个线程的一个补充… 下面是pahlevan发布的代码版本，但针对c# /XNA，并做了一些整理:

    /// <summary>
    /// Intersects a line and a circle.
    /// </summary>
    /// <param name="location">the location of the circle</param>
    /// <param name="radius">the radius of the circle</param>
    /// <param name="lineFrom">the starting point of the line</param>
    /// <param name="lineTo">the ending point of the line</param>
    /// <returns>true if the line and circle intersect each other</returns>
    public static bool IntersectLineCircle(Vector2 location, float radius, Vector2 lineFrom, Vector2 lineTo)
    {
        float ab2, acab, h2;
        Vector2 ac = location - lineFrom;
        Vector2 ab = lineTo - lineFrom;
        Vector2.Dot(ref ab, ref ab, out ab2);
        Vector2.Dot(ref ac, ref ab, out acab);
        float t = acab / ab2;

        if (t < 0)
            t = 0;
        else if (t > 1)
            t = 1;

        Vector2 h = ((ab * t) + lineFrom) - location;
        Vector2.Dot(ref h, ref h, out h2);

        return (h2 <= (radius * radius));
    }

我会用这个算法来计算点(圆心)和线(线AB)之间的距离。这可以用来确定直线与圆的交点。

假设有点A B c, Ax和Ay是A点的x和y分量。B和c也是一样，标量R是圆半径。

该算法要求A B C是不同的点，且R不为0。

这是算法

// compute the euclidean distance between A and B
LAB = sqrt( (Bx-Ax)²+(By-Ay)² )

// compute the direction vector D from A to B
Dx = (Bx-Ax)/LAB
Dy = (By-Ay)/LAB

// the equation of the line AB is x = Dx*t + Ax, y = Dy*t + Ay with 0 <= t <= LAB.

// compute the distance between the points A and E, where
// E is the point of AB closest the circle center (Cx, Cy)
t = Dx*(Cx-Ax) + Dy*(Cy-Ay)    

// compute the coordinates of the point E
Ex = t*Dx+Ax
Ey = t*Dy+Ay

// compute the euclidean distance between E and C
LEC = sqrt((Ex-Cx)²+(Ey-Cy)²)

// test if the line intersects the circle
if( LEC < R )
{
    // compute distance from t to circle intersection point
    dt = sqrt( R² - LEC²)

    // compute first intersection point
    Fx = (t-dt)*Dx + Ax
    Fy = (t-dt)*Dy + Ay

    // compute second intersection point
    Gx = (t+dt)*Dx + Ax
    Gy = (t+dt)*Dy + Ay
}

// else test if the line is tangent to circle
else if( LEC == R )
    // tangent point to circle is E

else
    // line doesn't touch circle