另一种方法使用三角形ABC面积公式。交点检验比投影法简单高效，但求交点坐标需要更多的工作。至少它会被推迟到需要的时候。

三角形面积的计算公式为:area = bh/2

b是底长，h是高。我们选择线段AB作为底，使h是圆心C到直线的最短距离。

因为三角形的面积也可以用向量点积来计算，所以我们可以确定h。

// compute the triangle area times 2 (area = area2/2)
area2 = abs( (Bx-Ax)*(Cy-Ay) - (Cx-Ax)(By-Ay) )

// compute the AB segment length
LAB = sqrt( (Bx-Ax)² + (By-Ay)² )

// compute the triangle height
h = area2/LAB

// if the line intersects the circle
if( h < R )
{
    ...
}        

更新1:

您可以通过使用这里描述的快速平方根倒数计算来优化代码，以获得1/LAB的良好近似值。

计算交点并不难。开始了

// compute the line AB direction vector components
Dx = (Bx-Ax)/LAB
Dy = (By-Ay)/LAB

// compute the distance from A toward B of closest point to C
t = Dx*(Cx-Ax) + Dy*(Cy-Ay)

// t should be equal to sqrt( (Cx-Ax)² + (Cy-Ay)² - h² )

// compute the intersection point distance from t
dt = sqrt( R² - h² )

// compute first intersection point coordinate
Ex = Ax + (t-dt)*Dx
Ey = Ay + (t-dt)*Dy

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

如果h = R，则直线AB与圆相切，且值dt = 0, E = F。点的坐标为E和F的坐标。

如果在应用程序中出现这种情况，您应该检查A与B是否不同，并且段长度不为空。

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;
        }

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

以下是我在TypeScript中的解决方案，遵循@Mizipzor建议的想法(使用投影):

/**
 * Determines whether a line segment defined by a start and end point intersects with a sphere defined by a center point and a radius
 * @param a the start point of the line segment
 * @param b the end point of the line segment
 * @param c the center point of the sphere
 * @param r the radius of the sphere
 */
export function lineSphereIntersects(
  a: IPoint,
  b: IPoint,
  c: IPoint,
  r: number
): boolean {
  // find the three sides of the triangle formed by the three points
  const ab: number = distance(a, b);
  const ac: number = distance(a, c);
  const bc: number = distance(b, c);

  // check to see if either ends of the line segment are inside of the sphere
  if (ac < r || bc < r) {
    return true;
  }

  // find the angle between the line segment and the center of the sphere
  const numerator: number = Math.pow(ac, 2) + Math.pow(ab, 2) - Math.pow(bc, 2);
  const denominator: number = 2 * ac * ab;
  const cab: number = Math.acos(numerator / denominator);

  // find the distance from the center of the sphere and the line segment
  const cd: number = Math.sin(cab) * ac;

  // if the radius is at least as long as the distance between the center and the line
  if (r >= cd) {
    // find the distance between the line start and the point on the line closest to
    // the center of the sphere
    const ad: number = Math.cos(cab) * ac;
    // intersection occurs when the point on the line closest to the sphere center is
    // no further away than the end of the line
    return ad <= ab;
  }
  return false;
}

export function distance(a: IPoint, b: IPoint): number {
  return Math.sqrt(
    Math.pow(b.z - a.z, 2) + Math.pow(b.y - a.y, 2) + Math.pow(b.x - a.x, 2)
  );
}

export interface IPoint {
  x: number;
  y: number;
  z: number;
}

这是一个Javascript实现。我的方法是首先将线段转换成一条无限的直线，然后找到交点。从那里，我检查是否找到的点在线段上。代码有良好的文档记录，您应该能够跟随。

您可以在这个现场演示中试用代码。 代码是从我的算法仓库里拿的。

// Small epsilon value
var EPS = 0.0000001;

// point (x, y)
function Point(x, y) {
  this.x = x;
  this.y = y;
}

// Circle with center at (x,y) and radius r
function Circle(x, y, r) {
  this.x = x;
  this.y = y;
  this.r = r;
}

// A line segment (x1, y1), (x2, y2)
function LineSegment(x1, y1, x2, y2) {
  var d = Math.sqrt( (x1-x2)*(x1-x2) + (y1-y2)*(y1-y2) );
  if (d < EPS) throw 'A point is not a line segment';
  this.x1 = x1; this.y1 = y1;
  this.x2 = x2; this.y2 = y2;
}

// An infinite line defined as: ax + by = c
function Line(a, b, c) {
  this.a = a; this.b = b; this.c = c;
  // Normalize line for good measure
  if (Math.abs(b) < EPS) {
    c /= a; a = 1; b = 0;
  } else { 
    a = (Math.abs(a) < EPS) ? 0 : a / b;
    c /= b; b = 1; 
  }
}

// Given a line in standard form: ax + by = c and a circle with 
// a center at (x,y) with radius r this method finds the intersection
// of the line and the circle (if any). 
function circleLineIntersection(circle, line) {

  var a = line.a, b = line.b, c = line.c;
  var x = circle.x, y = circle.y, r = circle.r;

  // Solve for the variable x with the formulas: ax + by = c (equation of line)
  // and (x-X)^2 + (y-Y)^2 = r^2 (equation of circle where X,Y are known) and expand to obtain quadratic:
  // (a^2 + b^2)x^2 + (2abY - 2ac + - 2b^2X)x + (b^2X^2 + b^2Y^2 - 2bcY + c^2 - b^2r^2) = 0
  // Then use quadratic formula X = (-b +- sqrt(a^2 - 4ac))/2a to find the 
  // roots of the equation (if they exist) and this will tell us the intersection points

  // In general a quadratic is written as: Ax^2 + Bx + C = 0
  // (a^2 + b^2)x^2 + (2abY - 2ac + - 2b^2X)x + (b^2X^2 + b^2Y^2 - 2bcY + c^2 - b^2r^2) = 0
  var A = a*a + b*b;
  var B = 2*a*b*y - 2*a*c - 2*b*b*x;
  var C = b*b*x*x + b*b*y*y - 2*b*c*y + c*c - b*b*r*r;

  // Use quadratic formula x = (-b +- sqrt(a^2 - 4ac))/2a to find the 
  // roots of the equation (if they exist).

  var D = B*B - 4*A*C;
  var x1,y1,x2,y2;

  // Handle vertical line case with b = 0
  if (Math.abs(b) < EPS) {

    // Line equation is ax + by = c, but b = 0, so x = c/a
    x1 = c/a;

    // No intersection
    if (Math.abs(x-x1) > r) return [];

    // Vertical line is tangent to circle
    if (Math.abs((x1-r)-x) < EPS || Math.abs((x1+r)-x) < EPS)
      return [new Point(x1, y)];

    var dx = Math.abs(x1 - x);
    var dy = Math.sqrt(r*r-dx*dx);

    // Vertical line cuts through circle
    return [
      new Point(x1,y+dy),
      new Point(x1,y-dy)
    ];

  // Line is tangent to circle
  } else if (Math.abs(D) < EPS) {

    x1 = -B/(2*A);
    y1 = (c - a*x1)/b;

    return [new Point(x1,y1)];

  // No intersection
  } else if (D < 0) {

    return [];

  } else {

    D = Math.sqrt(D);

    x1 = (-B+D)/(2*A);
    y1 = (c - a*x1)/b;

    x2 = (-B-D)/(2*A);
    y2 = (c - a*x2)/b;

    return [
      new Point(x1, y1),
      new Point(x2, y2)
    ];

  }

}

// Converts a line segment to a line in general form
function segmentToGeneralForm(x1,y1,x2,y2) {
  var a = y1 - y2;
  var b = x2 - x1;
  var c = x2*y1 - x1*y2;
  return new Line(a,b,c);
}

// Checks if a point 'pt' is inside the rect defined by (x1,y1), (x2,y2)
function pointInRectangle(pt,x1,y1,x2,y2) {
  var x = Math.min(x1,x2), X = Math.max(x1,x2);
  var y = Math.min(y1,y2), Y = Math.max(y1,y2);
  return x - EPS <= pt.x && pt.x <= X + EPS &&
         y - EPS <= pt.y && pt.y <= Y + EPS;
}

// Finds the intersection(s) of a line segment and a circle
function lineSegmentCircleIntersection(segment, circle) {

  var x1 = segment.x1, y1 = segment.y1, x2 = segment.x2, y2 = segment.y2;
  var line = segmentToGeneralForm(x1,y1,x2,y2);
  var pts = circleLineIntersection(circle, line);

  // No intersection
  if (pts.length === 0) return [];

  var pt1 = pts[0];
  var includePt1 = pointInRectangle(pt1,x1,y1,x2,y2);

  // Check for unique intersection
  if (pts.length === 1) {
    if (includePt1) return [pt1];
    return [];
  }

  var pt2 = pts[1];
  var includePt2 = pointInRectangle(pt2,x1,y1,x2,y2);

  // Check for remaining intersections
  if (includePt1 && includePt2) return [pt1, pt2];
  if (includePt1) return [pt1];
  if (includePt2) return [pt2];
  return [];

}

我根据chmike给出的答案为iOS创建了这个函数

+ (NSArray *)intersectionPointsOfCircleWithCenter:(CGPoint)center withRadius:(float)radius toLinePoint1:(CGPoint)p1 andLinePoint2:(CGPoint)p2
{
    NSMutableArray *intersectionPoints = [NSMutableArray array];

    float Ax = p1.x;
    float Ay = p1.y;
    float Bx = p2.x;
    float By = p2.y;
    float Cx = center.x;
    float Cy = center.y;
    float R = radius;


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

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

    // Now the line equation is x = Dx*t + Ax, y = Dy*t + Ay with 0 <= t <= 1.

    // compute the value t of the closest point to the circle center (Cx, Cy)
    float t = Dx*(Cx-Ax) + Dy*(Cy-Ay);

    // This is the projection of C on the line from A to B.

    // compute the coordinates of the point E on line and closest to C
    float Ex = t*Dx+Ax;
    float Ey = t*Dy+Ay;

    // compute the euclidean distance from E to C
    float LEC = sqrt( pow(Ex-Cx, 2)+ pow(Ey-Cy, 2) );

    // test if the line intersects the circle
    if( LEC < R )
    {
        // compute distance from t to circle intersection point
        float dt = sqrt( pow(R, 2) - pow(LEC,2) );

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

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

        [intersectionPoints addObject:[NSValue valueWithCGPoint:CGPointMake(Fx, Fy)]];
        [intersectionPoints addObject:[NSValue valueWithCGPoint:CGPointMake(Gx, Gy)]];
    }

    // else test if the line is tangent to circle
    else if( LEC == R ) {
        // tangent point to circle is E
        [intersectionPoints addObject:[NSValue valueWithCGPoint:CGPointMake(Ex, Ey)]];
    }
    else {
        // line doesn't touch circle
    }

    return intersectionPoints;
}

You can find a point on a infinite line that is nearest to circle center by projecting vector AC onto vector AB. Calculate the distance between that point and circle center. If it is greater that R, there is no intersection. If the distance is equal to R, line is a tangent of the circle and the point nearest to circle center is actually the intersection point. If distance less that R, then there are 2 intersection points. They lie at the same distance from the point nearest to circle center. That distance can easily be calculated using Pythagorean theorem. Here's algorithm in pseudocode:

{
dX = bX - aX;
dY = bY - aY;
if ((dX == 0) && (dY == 0))
  {
  // A and B are the same points, no way to calculate intersection
  return;
  }

dl = (dX * dX + dY * dY);
t = ((cX - aX) * dX + (cY - aY) * dY) / dl;

// point on a line nearest to circle center
nearestX = aX + t * dX;
nearestY = aY + t * dY;

dist = point_dist(nearestX, nearestY, cX, cY);

if (dist == R)
  {
  // line segment touches circle; one intersection point
  iX = nearestX;
  iY = nearestY;

  if (t < 0 || t > 1)
    {
    // intersection point is not actually within line segment
    }
  }
else if (dist < R)
  {
  // two possible intersection points

  dt = sqrt(R * R - dist * dist) / sqrt(dl);

  // intersection point nearest to A
  t1 = t - dt;
  i1X = aX + t1 * dX;
  i1Y = aY + t1 * dY;
  if (t1 < 0 || t1 > 1)
    {
    // intersection point is not actually within line segment
    }

  // intersection point farthest from A
  t2 = t + dt;
  i2X = aX + t2 * dX;
  i2Y = aY + t2 * dY;
  if (t2 < 0 || t2 > 1)
    {
    // intersection point is not actually within line segment
    }
  }
else
  {
  // no intersection
  }
}

编辑:增加了代码来检查所找到的交点是否实际上在线段内。