如何确定4个点是否在同一平面上

Question

我从Kinect输出的图像中选择了4个点,因此每个点都有其(x, y, z)坐标.

我的目标是确定4点是否落在同一平面上.

这是我的功能:

    public bool isValidPlane()
    {
        for (int i = 0; i < edgesPoints.Length; i++)
        {
            double absPlaneEquation = Math.Abs(distance -
                (normal.X * edgesPoints[i].X + normal.Y * edgesPoints[i].Y + normal.Z * edgesPoints[i].Z));
            if (absPlaneEquation > 1500) /* 1500 is a tolerance error*/
            {
                return false;
            }
        }
        return true;
    }

这normal也是法线(平面上2个向量的交叉乘积,之前已经从4个选定点中的3个计算得到)到平面并且它被标准化:

    private void calcPlaneNormalVector()
    {
        if (lastEdgeNumber < 3)
        {
            return;
        }
        Vector3D vec1 = new Vector3D(edgesPoints[0], edgesPoints[1]);
        Vector3D vec2 = new Vector3D(edgesPoints[0], edgesPoints[2]);
        vec2 = vec1.crossProduct(vec2);
        double lengthNormal = Math.Sqrt(Math.Pow(vec2.X, 2) + Math.Pow(vec2.Y, 2) + Math.Pow(vec2.Z, 2));
//normalizing:
        normal = new Vector3D((vec2.X / lengthNormal), (vec2.Y / lengthNormal), (vec2.Z / lengthNormal));
        distance = (-1) * (edgesPoints[0].X * normal.X + edgesPoints[0].Y * normal.Y + edgesPoints[0].Z + normal.Z);
    }

Vector3D 是一个表示向量的类:

public  class Vector3D
{
    private double x, y, z;

    public Vector3D(Point3D p1, Point3D p2)
    {
        x = p2.X - p1.X;
        y = p2.Y - p1.Y;
        z = p2.Z - p1.Z;
    }
    public Vector3D(double a = 0, double b = 0, double c = 0)
    {
        x = a;
        y = b;
        z = c;
    }

    <get properties for x, y, z >

    public Vector3D crossProduct(Vector3D u)
    {
        double tmpX = 0, tmpY = 0, tmpZ = 0;
        tmpX = y * u.Z - z * u.Y;
        tmpY = z * u.X - x * u.Z;
        tmpZ = x * u.Y - y * u.X;
        return new Vector3D(tmpX, tmpY, tmpZ);
    }
    public double dotProduct(Vector3D u)
    {
        return x * u.X + y * u.Y + z * u.Z;
    }
}

1300 <= absPlaneEquation <= 1400即使选择了4个点,我也总能得到它们,以便它们不会在同一个平面上.

检测4个点指向同一平面的最佳方法是什么？

Answer 1

获得平面的法向量后,可以计算平面的等式:

normal vector components : [A, B, C]
Plane equation           : A·x + B·y + C·z + D = 0;

使用三个点(P1,P2或P3)中的一个来获得法线向量来评估D,然后简单地检查第四个点(P4)是否满足等式:

 D = - (A·x1 + B·y1 + C·z1)
 A·x4 + B·y4 + C·z4 - (A·x1 + B·y1 + C·z1) = 0

值得注意的是,您正在使用浮点算术,因此您无法测试严格的相等性.您需要定义一个可接受的错误,并根据这样的容差检查第四个点是否符合等式:

 |A·x4 + B·y4 + C·z4 - (A·x1 + B·y1 + C·z1)| < TOLERANCE

更新:以下是我为您的问题编写解决方案的代码:

public struct Point3D
{
    public double X { get; }
    public double Y { get; }
    public double Z { get; }

    public Point3D(double x, double y, double z)
    {
        X = x;
        Y = y;
        Z = z;
    }
}

public struct Vector3D
{
    public double X { get; }
    public double Y { get; }
    public double Z { get; }
    public double Magnitude => Math.Sqrt(X * X + Y * Y + Z * Z);

    public Vector3D(Point3D p1, Point3D p2)
        : this(p2.X - p1.X, p2.Y - p1.Y, p2.Z - p1.Z)
    {
    }

    public Vector3D(double x, double y, double z)
    {
        X = x;
        Y = y;
        Z = z;
    }

    public static Vector3D CrossProduct(Vector3D left, Vector3D right)
    {
        double tmpX = 0, tmpY = 0, tmpZ = 0;
        tmpX = left.Y * right.Z - left.Z * right.Y;
        tmpY = left.Z * right.X - left.X * right.Z;
        tmpZ = left.X * right.Y - left.Y * right.X;
        return new Vector3D(tmpX, tmpY, tmpZ);
    }

    public static double DotProduct(Vector3D left, Vector3D right)
    {
        return left.X * right.X + left.Y * right.Y + left.Z * right.Z;
    }
}

public struct Plane3D
{
    private const double TOLERANCE = 0.001;

    private readonly double independentTerm;
    public Vector3D Normal { get; }

    public Plane3D(Point3D p1, Point3D p2, Point3D p3)
    {
        Normal = Vector3D.crossProduct(new Vector3D(p1, p2), new Vector3D(p1, p3));

        if (Normal.Magnitude < TOLERANCE)
            throw new ArgumentException("Specified points do not define a valid plane.");

        independentTerm = -(Normal.X * p1.X + Normal.Y * p1.Y + Normal.Z * p1.Z);
    }

    public bool Contains(Point3D p) => Math.Abs(Normal.X * p.X + Normal.Y * p.Y + Normal.Z * p.Z + independentTerm) < TOLERANCE;
}

注意事项:

1. 我已经改变了Point3D,并Vector3D以结构.这在很大程度上取决于您将如何使用这些对象,但乍一看,值类型似乎更合适.
2. 我已经使值类型不可变.可变值类型不是一个好主意; 再次,如果你将它们实现为类,这不会是一个问题,尽管我仍然建议尽可能创建不可变类型.在这种情况下,这样做非常便宜.
3. 你有飞机的概念,那么,创建一个代表一个的类型.
4. 我已经将向量运算符更改为static方法.这可能取决于个人品味.
5. 我已经在TOLERANCE里面实现了const Plane.可能有更好的地方来定义它,它只是为了方便.
6. 我推荐你稍微命名; 公众成员应以大写字母开头.