Sphere-Sphere相交c#，碰撞点的3D坐标

Question

我查看了许多 3D 球体-球体相交问题，不幸的是，它们要么超出我的理解能力，要么不适合我正在寻找的内容。

这是在 Unity 游戏引擎中并使用 c#

我已经设法让这段代码工作了：

public void calculatePoints_H(Vector3 c1p, Vector3 c2p, float c1r, float c2r, out Vector3 startLine, out Vector3 endLine)
{
    //c1p = circle one position
    //c1r = circle one radius

    Vector3 P0 = c1p;
    Vector3 P1 = c2p;

    float d,a,h;


    d = Vector3.Distance(P0,P1);

    a = (c1r*c1r - c2r*c2r + d*d)/(2*d);

    h = Mathf.Sqrt(c1r*c1r - a*a);


    Vector3 P2 = (P1 - P0);
            P2 = (P2 * (a/d));
            P2 = (P2 + P0);

    float x3,y3,x4,y4 = 0;

    x3 = P2.x + h*(P1.y - P0.y)/d;
    y3 = P2.y - h*(P1.x - P0.x)/d;

    x4 = P2.x - h*(P1.y - P0.y)/d;
    y4 = P2.y + h*(P1.x - P0.x)/d;;

    //draw visual to screen (unity 3D engine)
    Debug.DrawLine(new Vector3(x3,0,y3), new Vector3(x4,0,y4),Color.green);

    //out parameters for a line renderer
    startLine = new Vector3(x3,0,y3);
    endLine = new Vector3(x4,0,y4);


}

目前这段代码允许我计算两个球体相交的 x 轴和 z 轴上的两个点，然后画一条线。

我想要实现的是一个 xyz 交点，因此我还可以在方法中添加高度（y 向量 3 值），这样我就可以让一个球体从任何方向/高度与另一个球体相交

有人可以帮我理解如何解决这个问题吗，我的大脑有点炸了，我担心我缺少一个简单的解决方案？

Answer 1

球体很好，因为您知道交点（如果它们相接触）是沿矢量 AB 从 SphereA 中心到 SphereB 的一段距离> 中心。该距离是球体半径的函数：

float dA = SphereA.radius / AB.magnitude // % distance along AB starting from SphereA

从那里您可以计算出您的交点沿 AB 的距离：

Vector3 p = SphereA.position + AB * dA; // eq: p`= p + direction * time

示例（使用 Unity 的内置球体预制件）：

bool Intersect(out Vector3 ip, float threshold=0.1f){
    // vector from sphere 1 -> sphere 2
    Vector3 ab = Sphere2.transform.position - Sphere1.transform.position;

    // Calculate radius from Unity built-in sphere.
    // Unity spheres are unit spheres (diameter = 1)
    // So diameter = scale, thus radius = scale / 2.0f.
    // **Presumes uniform scaling.
    float r1 = Sphere1.transform.localScale.x / 2.0f;
    float r2 = Sphere2.transform.localScale.x / 2.0f;

    // When spheres are too close or too far apart, ignore intersection.
    float diff = Mathf.Abs(r2 + r1 - ab.magnitude);
    if( diff >= threshold) {
        ip = Vector3.zero;
        return false;
    }
    // Intersection is the distance along the vector between
    // the 2 spheres as a function of the sphere's radius.
    ip = Sphere1.transform.position + ab * r1/ab.magnitude;
    return true;
}

示例用法：

void FixedUpdate(){
    Vector3 p; //will hold intersection point if one is found
    if(Intersect(out p,0.1f)){
        IntersectionPoint.transform.position = p;
        IntersectionPoint.renderer.enabled = true;
    } else {
        IntersectionPoint.renderer.enabled = false;
    }
}

这只会返回一个交点。多个交点（例如球体重叠时）是需要解决的不同问题。