如果你找到了圆心(因为它是3D的，我想你是指球体而不是圆)和直线之间的距离，然后检查这个距离是否小于可以做到这一点的半径。

碰撞点显然是直线和球面之间最近的点(当你计算球面和直线之间的距离时，会计算出这个点)

点与线之间的距离: http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html

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;
}

基于@Joe Skeen的python解决方案

def check_line_segment_circle_intersection(line, point, radious):
    """ Checks whether a point intersects with a line defined by two points.

    A `point` is list with two values: [2, 3]

    A `line` is list with two points: [point1, point2]

    """
    line_distance = distance(line[0], line[1])
    distance_start_to_point = distance(line[0], point)
    distance_end_to_point = distance(line[1], point)

    if (distance_start_to_point <= radious or distance_end_to_point <= radious):
        return True

    # angle between line and point with law of cosines
    numerator = (math.pow(distance_start_to_point, 2)
                 + math.pow(line_distance, 2)
                 - math.pow(distance_end_to_point, 2))
    denominator = 2 * distance_start_to_point * line_distance
    ratio = numerator / denominator
    ratio = ratio if ratio <= 1 else 1  # To account for float errors
    ratio = ratio if ratio >= -1 else -1  # To account for float errors
    angle = math.acos(ratio)

    # distance from the point to the line with sin projection
    distance_line_to_point = math.sin(angle) * distance_start_to_point

    if distance_line_to_point <= radious:
        point_projection_in_line = math.cos(angle) * distance_start_to_point
        # Intersection occurs whent the point projection in the line is less
        # than the line distance and positive
        return point_projection_in_line <= line_distance and point_projection_in_line >= 0
    return False

def distance(point1, point2):
    return math.sqrt(
        math.pow(point1[1] - point2[1], 2) +
        math.pow(point1[0] - point2[0], 2)
    )

另一种解决方案，首先考虑不关心碰撞位置的情况。请注意，这个特定的函数是在假设xB和yB为向量输入的情况下构建的，但如果情况并非如此，则可以轻松修改。变量名在函数的开头定义

#Line segment points (A0, Af) defined by xA0, yA0, xAf, yAf; circle center denoted by xB, yB; rB=radius of circle, rA = radius of point (set to zero for your application)
def staticCollision_f(xA0, yA0, xAf, yAf, rA, xB, yB, rB): #note potential speed up here by casting all variables to same type and/or using Cython
    
    #Build equations of a line for linear agents (convert y = mx + b to ax + by + c = 0 means that a = -m, b = 1, c = -b
    m_v = (yAf - yA0) / (xAf - xA0)
    b_v = yAf - m_v * xAf
    rEff = rA + rB #radii are added since we are considering the agent path as a thin line

    #Check if points (circles) are within line segment (find center of line segment and check if circle is within radius of this point)
    segmentMask = np.sqrt( (yB - (yA0+yAf)/2)**2 + (xB - (xA0+xAf)/2)**2 ) < np.sqrt( (yAf - yA0)**2 + (xAf - xA0)**2 ) / 2 + rEff

    #Calculate perpendicular distance between line and a point
    dist_v = np.abs(-m_v * xB + yB - b_v) / np.sqrt(m_v**2 + 1)
    collisionMask = (dist_v < rEff) & segmentMask

    #return True if collision is detected
    return collisionMask, collisionMask.any()

如果您需要碰撞的位置，您可以使用这个站点上详细介绍的方法，并将其中一个代理的速度设置为零。这种方法也适用于矢量输入:http://twobitcoder.blogspot.com/2010/04/circle-collision-detection.html

采取

E是射线的起点， L是射线的端点， C是你测试的圆心 R是球面的半径

计算: d = L - E(射线方向矢量，从头到尾) f = E - C(从中心球到射线起点的向量)

然后通过…找到交点。 堵塞: P = E + t * d 这是一个参数方程 Px = Ex + tdx Py = Ey + tdy 成 (x - h)2 + (y - k)2 = r2 (h,k) =圆心。

注意:我们在这里将问题简化为2D，我们得到的解决方案也适用于3D

得到:

Expand x2 - 2xh + h2 + y2 - 2yk + k2 - r2 = 0 Plug x = ex + tdx y = ey + tdy ( ex + tdx )2 - 2( ex + tdx )h + h2 + ( ey + tdy )2 - 2( ey + tdy )k + k2 - r2 = 0 Explode ex2 + 2extdx + t2dx2 - 2exh - 2tdxh + h2 + ey2 + 2eytdy + t2dy2 - 2eyk - 2tdyk + k2 - r2 = 0 Group t2( dx2 + dy2 ) + 2t( exdx + eydy - dxh - dyk ) + ex2 + ey2 - 2exh - 2eyk + h2 + k2 - r2 = 0 Finally, t2( d · d ) + 2t( e · d - d · c ) + e · e - 2( e · c ) + c · c - r2 = 0 Where d is the vector d and · is the dot product. And then, t2( d · d ) + 2t( d · ( e - c ) ) + ( e - c ) · ( e - c ) - r2 = 0 Letting f = e - c t2( d · d ) + 2t( d · f ) + f · f - r2 = 0

所以我们得到: T2 *(d·d) + 2t*(f·d) + (f·f - r2) = 0

求解二次方程:

float a = d.Dot( d ) ;
float b = 2*f.Dot( d ) ;
float c = f.Dot( f ) - r*r ;

float discriminant = b*b-4*a*c;
if( discriminant < 0 )
{
  // no intersection
}
else
{
  // ray didn't totally miss sphere,
  // so there is a solution to
  // the equation.
  
  discriminant = sqrt( discriminant );

  // either solution may be on or off the ray so need to test both
  // t1 is always the smaller value, because BOTH discriminant and
  // a are nonnegative.
  float t1 = (-b - discriminant)/(2*a);
  float t2 = (-b + discriminant)/(2*a);

  // 3x HIT cases:
  //          -o->             --|-->  |            |  --|->
  // Impale(t1 hit,t2 hit), Poke(t1 hit,t2>1), ExitWound(t1<0, t2 hit), 

  // 3x MISS cases:
  //       ->  o                     o ->              | -> |
  // FallShort (t1>1,t2>1), Past (t1<0,t2<0), CompletelyInside(t1<0, t2>1)
  
  if( t1 >= 0 && t1 <= 1 )
  {
    // t1 is the intersection, and it's closer than t2
    // (since t1 uses -b - discriminant)
    // Impale, Poke
    return true ;
  }

  // here t1 didn't intersect so we are either started
  // inside the sphere or completely past it
  if( t2 >= 0 && t2 <= 1 )
  {
    // ExitWound
    return true ;
  }
  
  // no intn: FallShort, Past, CompletelyInside
  return false ;
}

也许有另一种方法来解决这个问题，使用坐标系的旋转。

通常，如果一个线段是水平的或垂直的，这意味着平行于x轴或y轴，交点的求解很容易，因为我们已经知道交点的一个坐标，如果有的话。剩下的显然是用圆的方程找到另一个坐标。

受此启发，我们可以利用坐标系旋转，使一个轴的方向与线段的方向重合。

让我们以圆x^2+y^2=1和线段P1-P2为例，P1(-1.5,0.5)和P2(-0.5，-0.5)在x-y系统中。下面的方程提醒你旋转的原理，其中是逆时针方向的角度，x'-y'是旋转后的方程组:

x'=x*cos () + y*sin () y' = - x*sin () + y*cos ()

和反向

X = X ' * cos - y' * sin Y = x' * sin + Y ' * cos

考虑P1-P2方向(用-x表示为45°)，我们可以取=45°。将第二个旋转方程转化为x-y系统中的圆方程:x^2+y^2=1，经过简单的运算，我们得到x'-y'系统中的“相同”方程:x'^2+y'^2=1。

利用第一个旋转方程=> P1(-根号(2)/2，根号(2))，P2(-根号(2)/ 2,0)，线段端点变成x'-y'系统。

假设交点为p，在x'-y'中，Px = -根号2 /2。使用新的圆方程，我们得到Py = +根号(2)/2。将P转换成原始的x-y系统，最终得到P(-1,0)

为了实现这个数值，我们可以先看看线段的方向:水平，垂直或不垂直。如果它属于前两种情况，很简单。如果是最后一种情况，应用上述算法。

为了判断是否有交集，我们可以将解与端点坐标进行比较，看看它们之间是否有一个根。

我相信只要我们有了它的方程，这个方法也可以应用于其他曲线。唯一的缺点是，我们应该在x'-y'坐标系下解方程，这可能很难。