我如何确定3维矢量是否包含在由其他3个矢量形成的锐化区域内? [英] How do I determine whether a 3-dimensional vector is contained within the acute region formed by three other vectors?

问题描述

我正在C#中的一个项目中，在R3中有三个向量，我需要确定在这些向量形成的区域内是否包含第三个向量.这三个基向量之间的任何两个之间的最大角度均为90度，并且都在单位球面上进行了归一化.它们可以是负数.

I'm working on a project in C# where I have three vectors in R3, and I need to find out if a third vector is contained within the region formed by those vectors. The three basis vectors have a maximum angle of 90 degrees between any two of them, and they are all normalized on the unit sphere. They can be negative.

到目前为止，我已经尝试使用矩阵向量乘法来查找向量的变换坐标.从那里，我检查所有三个分量是否都为正.

So far, I've tried matrix-vector multiplication to find the transformed coordinates of the vector. From there, I check whether all three components are positive.

此方法在大多数情况下都有效，但是在处理特定的向量集时会遇到一些问题.问题似乎来自基向量的顺序.基本向量没有特定的顺序进行处理.我需要一种以(x，y，z)对这三个基础向量进行排序的方法，以确保变换将目标向量正确"映射到空间的正区域时，将其正确地映射到这些向量的正"内部，并将其正确映射到在外面"的空间.

This method works in most cases, but it has some issues working with specific sets of vectors. The issue seems to come from the order of the basis vectors. The basis vectors are processed in no specific order. I need a way to sort these three basis vectors in (x, y, z) order to ensure that the transformation correctly maps my target vector into the positive region of space when it is "inside" these vectors, and into a negative region of space when it is "outside".

非常感谢所有帮助.

2D相关讨论:如何确定一个向量是否在其他两个向量之间?

这些是我最终使用的解决方案.第一个与Edward的解决方案非常接近，第二个为了我的项目的一致性而稍作更改.

These are the solutions I ended up using. The first is very close to Edward's solution, and the second is slightly changed for consistency in my project.

public bool Vec3Between(Vector3 s, Vector3 p, Vector3 q, Vector3 r)
{
    Vector3 cross = Vector3.Cross(q, r);
    float ds = cross.x * s.x + cross.y * s.y + cross.z * s.z;
    float dp = cross.x * p.x + cross.y * p.y + cross.z * p.z;
    bool same_qr = (ds * dp >= 0);

    cross = Vector3.Cross(r, p);
    ds = cross.x * s.x + cross.y * s.y + cross.z * s.z;
    float dq = cross.x * q.x + cross.y * q.y + cross.z * q.z;
    bool same_rp = (ds * dq >= 0);

    cross = Vector3.Cross(p, q);
    ds = cross.x * s.x + cross.y * s.y + cross.z * s.z;
    float dr = cross.x * r.x + cross.y * r.y + cross.z * r.z;
    bool same_pq = (ds * dr >= 0);

    return same_qr && same_rp && same_pq;
}

public bool Vec3Between(Vector3 vTarget, Vector3[] vRef)
{
    bool same_side = true;
    for (int i = 0; i < 3; i++)
    {
        int i1 = (i < 2) ? (i + 1) : 0;
        int i2 = (i > 0) ? (i - 1) : 2;
        Vector3 cross = Vector3.Cross(vRef[i1], vRef[i2]);
        float plane_vTarget = cross.x * vTarget.x + cross.y * vTarget.y + cross.z * vTarget.z;
        float plane_vRef = cross.x * vRef[i].x + cross.y * vRef[i].y + cross.z * vRef[i].z;
        same_side = same_side && (plane_vTarget * plane_vRef >= 0);
    }
    return same_side;
}

推荐答案

如果您发布了一些示例，也许还有一些代码，那就太好了.

It would be nice if you posted some examples, and maybe some code.

在没有该信息的情况下，让我提出另一种方法.让定义空间区域的三个向量为p=(p1,p2,p3)，q=(q1,q2,q3)，r=(r1,r2,r3)和第四点s=(s1,s2,s3).我们需要执行3个测试:

In the absence of that information, let me suggest another approach. Let the three vectors defining the region of space be p=(p1,p2,p3), q=(q1,q2,q3), r=(r1,r2,r3), and the fourth point s=(s1,s2,s3). We need to perform 3 tests:

1)s与由q和r形成的平面的p在同一侧吗?

1) Is s on the same side as p of the plane formed by q and r?

2)s与由r和p形成的平面的q在同一侧吗?

2) Is s on the same side as q of the plane formed by r and p?

3)s与由p和q形成的平面的r在同一侧吗?

3) Is s on the same side as r of the plane formed by p and q?

所有三个问题的是"等同于您要寻找的几何特性.我将向您展示如何回答1).然后2)和3)是相似的.

A yes to all three questions is equivalent to the geometric property you are looking for. I'll show you how to answer 1). Then 2) and 3) are analogous.

我们通过取叉积q x r = (q2*r3-q3*r2, q3*r1-q1*r3, q1*r2-q2*r1) = (a,b,c)，找到通过q，r和原点的平面.则平面的等式为a*x + b*y + c*z = 0.

We find the plane through q, r and the origin by taking the cross product q x r = (q2*r3-q3*r2, q3*r1-q1*r3, q1*r2-q2*r1) = (a,b,c). Then the equation of the plane is a*x + b*y + c*z = 0.

我们将s的坐标放入上式的lhs中，得到表达式a*s1 + b*s2 + c*s3，它的值等于ds.我们将p的坐标放入上述公式的上述lhs中，得到表达式a*p1 + b*p2 + c*p3，该表达式具有一些值dp，例如.

We put the coordinates of s into the lhs of the above formula and get the expression a*s1 + b*s2 + c*s3, which has some value ds, say. We put the coordinates of p into the above lhs of the above formula and get the expression a*p1 + b*p2 + c*p3, which has some value dp, say.

如果ds和dp具有相同的符号，则s和p在平面的同一侧.如果ds和dp具有不同的符号，则s和p在平面的相对侧.

If ds and dp have the same sign, then s and p are on the same side of the plane. If ds and dp have different signs, then s and p are on opposite sides of the plane.

这是整个测试的布尔表达式:

Here's a boolean expression for the overall test:

( 
 ((q2*r3-q3*r2)*s1 + (q3*r1-q1*r3)*s2 + (q1*r2-q2*r1)*s3)
*((q2*r3-q3*r2)*p1 + (q3*r1-q1*r3)*p2 + (q1*r2-q2*r1)*p3)
> 0
)
&&
( 
 ((r2*p3-r3*p2)*s1 + (r3*r1-p1*r3)*s2 + (r1*p2-r2*p1)*s3)
*((r2*p3-r3*p2)*q1 + (r3*p1-r1*p3)*q2 + (r1*p2-r2*p1)*q3)
> 0
)
&&
( 
 ((p2*q3-p3*q2)*s1 + (p3*q1-p1*q3)*s2 + (p1*q2-p2*q1)*s3)
*((p2*q3-p3*q2)*r1 + (p3*q1-p1*q3)*r2 + (p1*q2-p2*q1)*r3)
> 0
)

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