直线的平面方程

Question

好吧，我需要做两件事：

1. 我需要确定 一条线，具有给定的角度，从 a 3D 空间中的点
2. 我需要确定方程 垂直于的平面 那条线，并且是一个设定的尺寸，与 原来的线在它的中心。

我需要平面方程的形式，给定一个新的直线方程，我可以知道它在平面上的相交位置（假设它首先相交）。

Answer 1

1. 那么 3D 空间中的点将采用从 X,Y,Z 原点 0,0,0 开始的 3D 向量的形式？

角度是相对于什么线的？

2. 叉积将为您提供垂直线。

    Function CrossProduct(ByVal b As Vector3d) As Vector3d
        'cross product = (ay*bz - az*by, az*bx - ax*bz, ax*by - ay*bx)
        Dim cp As New Vector3d
        cp.x = y * b.z - z * b.y
        cp.y = z * b.x - x * b.z
        cp.z = x * b.y - y * b.x
        Return cp
    End Function

    Function DotProduct(ByVal OtherVector As Vector3d) As Double
        'calculate dot product of two vectors
        Return x * OtherVector.x + y * OtherVector.y + z * OtherVector.z
    End Function

    Public Class Ray3d
        Public Po As New Vector3d    'point of origin
        Public V As New Vector3d    'vector
    End Class

    Public Class Plane3d
        Public N As New Vector3d    'normal
        Public PoP As New Vector3d   'point on plane
    End Class


    Private Function IntersectionTest(ByVal R As Ray3d, ByVal P As Plane3d, ByRef ReturnPoint As Vector3d) As Boolean

        Dim RayDotPlaneNormal As Double = R.V.DotProduct(P.N)

        If RayDotPlaneNormal = 0 for 1 sided
            Return False 'no intersection
        End If

        'PLANE EQUATION PoP.N = d

        Dim d As Double
        Dim PopVector As Vector3d = P.PoP.ToVector3d
        d = P.N.DotProduct(PopVector)


        'INTERSECTION EQUATION
        't = -(Po.N+d)/(V.N)

        Dim PointOriginVector As Vector3d
        PointOriginVector = R.Po.ToVector3d

        Dim PointOriginDotPlaneNormal As Double
        PointOriginDotPlaneNormal = P.N.DotProduct(PointOriginVector)

        Dim t As Double
        t = -(PointOriginDotPlaneNormal + d) / RayDotPlaneNormal

        ReturnPoint.x = R.Po.x + R.V.x * t
        ReturnPoint.y = R.Po.y + R.V.y * t
        ReturnPoint.z = R.Po.z + R.V.z * t

        Return True

    End Function