两线段交点

假设有如下图的两段线段BA, CD，求交点E:

求线段与线段交点

先求得C和D是否在向量BA两边，BA叉乘BC，BA叉乘BD，分别得到法向量两条法向量n1,n2，并单位化：
$\vec n1 = normalize(BA \times BC)$
$\vec n2 = normalize(BA \times BD)$

得到的n1和n2都垂直于BA和CD组成的平面，通过点乘求的两条法向量夹角，便可知道C和D是否在向量BA两侧:
$\vec n1 \cdot \vec n2 = \mid \vec n1 \mid \mid \vec n1 \mid cos\theta$

1. n1 x n2的结果如果大于0，锐角，说明C和D在向量BA同侧，BA和CD不可能相交。
$\vec n1 \cdot \vec n2 > 0$

2. n1 x n2的结果如果小于0，钝角，说明C和D在向量BA两侧，BA和CD可能相交。
$\vec n1 \cdot \vec n2 < 0$

3. 考虑到向量BA和CD有可能平行，BA和CD需要点乘求夹角，已证明是否平衡。

同理，我们求得A和B是否分别在向量CD两侧。假设最终结果是，C和D在向量BA两侧，并且A和B也在向量CD两侧，两线段有交点。接下来我们求交点E得坐标。作BA垂直线C'D'，以C为起点，CD投影C'D'，CA投影C'O。在梯形CC'D'D中，满足以下等式:
$\frac{C'O}{C'D'} = \frac{CE}{CD}$

C'O和C'D'长度已知，CD向量已知，因此可求得CE向量。

using UnityEngine;

public class Sample : MonoBehaviour
{
    private void Start()
    {
        //假设在XZ平面，忽略Y轴。
        Vector3 A = new Vector3(0, 0, 4);
        Vector3 B = new Vector3(0, 0, -1);
        Vector3 C = new Vector3(-2, 0, 1);
        Vector3 D = new Vector3(5, 0, 3);

        //尝试得到交点
        if (TryGetPointOfIntersection(B, A, C, D, out Vector3 resultPoint))
        {
            Debug.Log($"Sample.Start() : 交点为{resultPoint}");
        }
    }

    /// <summary>
    /// 尝试获取交点
    /// </summary>
    private bool TryGetPointOfIntersection(Vector3 fromPoint1, Vector3 toPoint1, Vector3 fromPoint2, Vector3 toPoint2, out Vector3 resultPoint)
    {
        if (!IsIntersect(fromPoint1, toPoint1, fromPoint2, toPoint2))
        {
            resultPoint = Vector3.zero;
            return false;
        }

        //两条向量
        Vector3 vector1 = toPoint1 - fromPoint1;
        Vector3 vector2 = toPoint2 - fromPoint2;

        //求向量1垂线.(必须单位化，这样点乘求的才是投影，你懂我意思吗)
        Vector3 perpendicularLine = new Vector3(vector1.z, 0, -vector1.x).normalized;

        //投影
        //相当于C'O
        float shadowVector1 = Vector3.Dot(toPoint1 - fromPoint2, perpendicularLine);

        //相当于C'D'
        float shadowVector2 = Vector3.Dot(vector2, perpendicularLine);

        //交点
        resultPoint = fromPoint2 + vector2 * (shadowVector1 / shadowVector2);

        return true;
    }

    /// <summary>
    /// 是否相交
    /// </summary>
    private bool IsIntersect(Vector3 fromPoint1, Vector3 toPoint1, Vector3 fromPoint2, Vector3 toPoint2)
    {
        return IsTwoSide(fromPoint1, toPoint1, fromPoint2, toPoint2) && IsTwoSide(fromPoint2, toPoint2, fromPoint1, toPoint1);
    }

    /// <summary>
    /// 两点是否在向量两侧
    /// </summary>
    private bool IsTwoSide(Vector3 fromPoint1, Vector3 toPoint1, Vector3 fromPoint2, Vector3 toPoint2)
    {
        Vector3 baseVector = toPoint1 - fromPoint1;
        Vector3 normal1 = Vector3.Cross(baseVector, fromPoint2 - fromPoint1).normalized;
        Vector3 normal2 = Vector3.Cross(baseVector, toPoint2 - fromPoint1).normalized;

        return Vector3.Dot(normal1, normal2) < 0;
    }
}

红色点为交点

Author : SunnyDecember
Date : 2019.11.17
原文

最后编辑于：2021.06.16 21:21:54

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