Rotated Star Tip

I sometimes use a single rotated star tip to measure 5th axis holes that would otherwise need a separate part setup. Auto Wrist Toggle does a good job for simple probe builds but ignores the adjustable nature of a star hub (rotational). One can nudge the tip around until it looks right but I grew tired of the trial by error method. I wanted to know what probe rotations along with star tip hub rotations might work for a given hole. After fooling around with Cartesian to Spherical conversions I ended up with the attached code. This is programmed in VBA and makes use of an Excel worksheet for output.

The probe head is assumed to be similar to a PH-10 having A & B rotations in the same manner.
The stylus is attached to the hub that coincides with a probe rotation of A90B90 (hub #2). This is the C axis 0° position.

---How to Use---
Modify the values in the USER DEFINED DATA section of the AllStarAngles procedure

PcDmis must be open with the active part program having a circle or cylinder auto feature selected in the Edit Window (cursor must be located within the feature).

Run the StartHere procedure.

Pick a resulting A-B-C combination that suits.

Open Probe Utilities and add the A-B angles. Double-click on the connection having the 5-way star cluster hub ("EXTEN5WAY" in my case). This displays the "Edit Probe Component" dialog where the "Default rotation angle about connection" can be edited. Enter the "C" angle.

With the newly defined tip selected in the program edit (F9) the hole feature. Use the View Normal Toggle to line things up. Turning off display of all but stylus will help.

The routine takes the vector of the selected feature and transforms it from the current alignment to the machine axis (Part Setup transforms are OK). Orientation of head mounting provides a rotation matrix (in machine axis). Orientation of sensor after A-B rotations provides another rotation matrix in a local system (the left-handedness is accommodated). The hole axis transformation is done using the PcDmis Object Library. The remaining math uses traditional methods.

The default values for head orientation and rotational limits reflect the setup I use for a bridge having a Tesastar-M with the LED facing Yminus.

I'm sure there are more efficient ways to cover this ground. I just picked a method that made sense to me. Error handling is minimal but I did document the code as best I could.

Code part 1 of 2

Option Explicit
Public Const PI As Double = 3.14159265358979

Sub StartHere()
    Call AllStarAngles
End Sub

Sub AllStarAngles()
    Dim Hole(2) As Double  'hole IJK outward axis direction
    Dim HoleInv(2) As Double  'hole axis inverted
    Dim HeadMat(2, 2) As Double 'probe head matrix in machine system
    Dim SensMat(2, 2) As Double 'probe sensor matrix in head system
    Dim Amin As Single, Amax As Single, Ainc As Single  'probe A rotations
    Dim Bmin As Single, Bmax As Single, Binc As Single  'probe B rotations
    Dim BallDia As Single, StemDia As Single  'stylus
    Dim a0b0 As String, a90b180 As String  'probe head orientations
    Dim A As Double, B As Double  'loop counters doubling as probe rotation angles
    Dim C As Double  'star tip 5-way rotation
    Dim StarErr As Double  'misalignment of star tip with hole in decimal degrees
    Dim CurRow As Integer  'Excel row
    Dim TipClr As Single  'stylus clearance between ball & stem
    Dim ShankDepth As Single
    
    On Error GoTo ErrCatcher
    
    Call GetPcDmisHole(Hole)
    If Hole(0) = -9 Then
        MsgBox ("Need a hole feature selected in Edit Window")
        GoTo ExitSub
    End If
    HoleInv(0) = -Hole(0): HoleInv(1) = -Hole(1): HoleInv(2) = -Hole(2)
    
    '_____USER DEFINED DATA_____
    'this block of 10 assignments should come from form
    a0b0 = "ZMINUS"
    a90b180 = "YPLUS"
    Amin = -115
    Amax = 90
    Ainc = 5
    Bmin = -180
    Bmax = 180
    Binc = 5
    BallDia = 0.1181
    StemDia = 0.0787
 
    TipClr = (BallDia - StemDia) / 2
    
    Call GetProbeHeadMatrix(a0b0, a90b180, HeadMat)
    
    'following 3 lines for Excel output
    ActiveWorkbook.Sheets("Scratch").Activate
    Sheets("Scratch").Rows("2:" & Rows.Count).ClearContents
    CurRow = 2
    
    For A = Amin To Amax Step Ainc
        For B = Bmin To Bmax Step Binc
            Call GetSensorMatrix(B, A, HeadMat, SensMat)
            StarErr = StarAngle(HoleInv, SensMat, C)  'StarErr = [COLOR="#0000FF"]angular misalignment[/COLOR]
            ShankDepth = TipClr / Sin(Radians(StarErr))
            Cells(CurRow, 1) = A  'probe A rotation
            Cells(CurRow, 2) = B  'probe B rotation
            Cells(CurRow, 3) = C  'star hub rotation
            Cells(CurRow, 4) = StarErr  'mis-alignment of tip & hole in decimal degrees
            Cells(CurRow, 5) = ShankDepth  'max probing depth based on StarErr mis-alignment & stylus data given
            CurRow = CurRow + 1  'increment worksheet row
        Next B
    Next A
    Columns("A:E").Sort key1:=Range("D2"), order1:=xlAscending, Header:=xlYes
    
ExitSub:
    Exit Sub
ErrCatcher:
    MsgBox ("Error: " & Err.Number & vbCrLf & "Description: " & Err.Description)
    On Error GoTo 0
    Resume ExitSub
End Sub

Private Sub GetPcDmisHole(H() As Double)
    'extract vector of currently selected hole or cylinder in Edit Window and convert to machine axis
    
    Dim oApp As PCDLRN.Application
    Dim oPart As PCDLRN.PartProgram
    Dim oCmds As PCDLRN.Commands
    Dim oCmd As PCDLRN.Command
    Dim oFeatCmd As PCDLRN.FeatCmd
    Dim oAlign As PCDLRN.AlignCmnd
    Dim oMatrix As PCDLRN.DmisMatrix
    Dim oPnt As PCDLRN.PointData
    Dim CmdType As Long
    Dim i As Double, j As Double, k As Double
    Dim bRet As Boolean
    Dim A(2, 2) As Double
    
    Set oApp = CreateObject("PCDLRN.Application")
    Set oPart = oApp.ActivePartProgram
    Set oCmds = oPart.Commands
    Set oCmd = oCmds.CurrentCommand
    
    CmdType = oCmd.Type
    If CmdType = 612 Or CmdType = 616 Then
        Set oFeatCmd = oCmd.FeatureCommand
        Set oCmd = oCmds.CurrentAlignment
        Set oAlign = oCmd.AlignmentCommand
        Set oMatrix = oAlign.MachineToPartMatrix
        Set oPnt = oMatrix.PrimaryAxis  'used to initialize PointData object - primary axis not used
        ' .GetVector does not take a PointData object
        bRet = oFeatCmd.GetVector(FVECTOR_SURFACE_VECTOR, FDATA_THEO, i, j, k)
        oPnt.i = i: oPnt.j = j: oPnt.k = k
        ' .TransformDataBack takes only a PointData object
        oMatrix.TransformDataBack oPnt, ROTATE_ONLY, PLANE_TOP
        H(0) = oPnt.i: H(1) = oPnt.j: H(2) = oPnt.k
    Else
        H(0) = -9  'current command in Edit Window not valid feature - flag error
    End If
    
    Set oPnt = Nothing
    Set oMatrix = Nothing
    Set oAlign = Nothing
    Set oFeatCmd = Nothing
    Set oCmd = Nothing
    Set oCmds = Nothing
    Set oPart = Nothing
    Set oApp = Nothing
End Sub

Private Function StarAngle(H() As Double, S() As Double, C As Double) As Double
    'H(2) = inward hole axis IJK
    'S(2,2) = sensor matrix
    'C = returned star tip rotation
    'StarAng returns smallest DD angular error using C rotation
    
    Dim V(2) As Double  'vector corresponding to star tip 0 angle
    Dim W(2) As Double  'vector upwards along sensor axis
    Dim HP(2) As Double  'hole axis projected
    Dim R(2) As Double  'rotated star tip vector (outward)
    Dim T(2) As Double  'temp vec
    
    Call PullMatRow(S, 1, V)  '5-way hub #2 - stylus default location
    Call PullMatRow(S, 2, W)  ' "Z" sensor axis running from sensor towards head
    
    Call ProjectVecOntoPlane(H, W, HP)  'project hole axis onto star hub rotational plane
    C = vDotAng(V, HP)  'angle to rotate star hub
    
    'determine if C is CW or CCW
    Call vCross(V, HP, T)
    If vDotAng(W, T) > 90 Then C = 360 - C
    
    Call RotVecAxis(V, W, C, R)  'R = vector of star tip rotated by C (pointing outward)
    StarAngle = vDotAng(R, H)  '3D angle between hole and rotated tip
End Function

Private Sub GetSensorMatrix(dZ As Double, dY As Double, Head() As Double, M() As Double)
    'populate M 3x3 matrix from probe head rotations given in decimal degrees
    'rotations in order Z-Y-X
    'dZ = RotB = about Z in XY plane
    'dY = RotA = about Y in ZX plane
    'Head(2,2) = probe head orientation for how mounted to CMM
    
    Dim sinZ As Double, sinY As Double, cosZ As Double, cosY As Double
    Dim T(2, 2) As Double
    
    'perform 1 time evaluation of trig functions
    sinZ = Sin(Radians(dZ)): sinY = Sin(Radians(dY))
    cosZ = Cos(Radians(dZ)): cosY = Cos(Radians(dY))
    
    T(0, 0) = cosZ * cosY
    T(0, 1) = sinZ * cosY
    T(0, 2) = sinY
    T(1, 0) = -sinZ
    T(1, 1) = cosZ
    T(1, 2) = 0
    T(2, 0) = -cosZ * sinY
    T(2, 1) = -sinZ * sinY
    T(2, 2) = cosY
    Call MatMult(T, Head, M)  'put in machine axis
End Sub

Parents

GomoFazter over 8 years ago

Code part 2 of 2

Private Sub GetProbeHeadMatrix(a0b0 As String, a90b180 As String, pMat() As Double)
    'determine 3x3 probe matrix given 2 initial probe rotation directions
    'creates frame to transform from head axis to machine axis
    'a0b0 & a90b180 represent directions pointed towards in PcDmis
    'allowed strings: XMINUS, XPLUS, YMINUS, YPLUS, ZMINUS, ZPLUS
    
    Dim Val As Integer, Axis As Integer
    
    'zero out rotation matrix
    pMat(0, 0) = 0: pMat(0, 1) = 0: pMat(0, 2) = 0
    pMat(2, 0) = 0: pMat(2, 1) = 0: pMat(2, 2) = 0
    
    'a0b0 for probe body Z axis
    Val = 1
    If UCase(Mid(a0b0, 2, 1)) = "P" Then Val = -Val  'invert direction
    Axis = Asc(UCase(Left$(a0b0, 1))) - 88
    pMat(2, Axis) = Val
    
    'a90b180 for probe body X axis
    Val = 1
    If UCase(Mid(a90b180, 2, 1)) = "P" Then Val = -Val  'invert direction
    Axis = Asc(UCase(Left$(a90b180, 1))) - 88
    pMat(0, Axis) = Val
    
    'cross product for probe body Y axis
    pMat(1, 0) = pMat(2, 1) * pMat(0, 2) - pMat(2, 2) * pMat(0, 1)
    pMat(1, 1) = pMat(2, 2) * pMat(0, 0) - pMat(2, 0) * pMat(0, 2)
    pMat(1, 2) = pMat(2, 0) * pMat(0, 1) - pMat(2, 1) * pMat(0, 0)
End Sub

Private Sub PullMatRow(M() As Double, Row As Integer, V() As Double)
    'extract a row from a matrix (dumb VBA)
    V(0) = M(Row, 0)
    V(1) = M(Row, 1)
    V(2) = M(Row, 2)
End Sub

Private Function ArcCos(X As Double) As Double
    'return the arc cosine
    ArcCos = 0
    If X < 1 Then
        If X > -1 Then
            ArcCos = Atn(-X / Sqr(-X * X + 1)) + 2 * Atn(1)
        Else
            ArcCos = 180
        End If
    End If
End Function

Private Sub MatMult(A() As Double, B() As Double, C() As Double)
    'matrix multiplication - takes 2 3x3 matrixes
    'a #columns = b #rows,  c = a #rows & b #columns
    C(0, 0) = A(0, 0) * B(0, 0) + A(0, 1) * B(1, 0) + A(0, 2) * B(2, 0)
    C(0, 1) = A(0, 0) * B(0, 1) + A(0, 1) * B(1, 1) + A(0, 2) * B(2, 1)
    C(0, 2) = A(0, 0) * B(0, 2) + A(0, 1) * B(1, 2) + A(0, 2) * B(2, 2)
    C(1, 0) = A(1, 0) * B(0, 0) + A(1, 1) * B(1, 0) + A(1, 2) * B(2, 0)
    C(1, 1) = A(1, 0) * B(0, 1) + A(1, 1) * B(1, 1) + A(1, 2) * B(2, 1)
    C(1, 2) = A(1, 0) * B(0, 2) + A(1, 1) * B(1, 2) + A(1, 2) * B(2, 2)
    C(2, 0) = A(2, 0) * B(0, 0) + A(2, 1) * B(1, 0) + A(2, 2) * B(2, 0)
    C(2, 1) = A(2, 0) * B(0, 1) + A(2, 1) * B(1, 1) + A(2, 2) * B(2, 1)
    C(2, 2) = A(2, 0) * B(0, 2) + A(2, 1) * B(1, 2) + A(2, 2) * B(2, 2)
End Sub

Private Sub ProjectVecOntoPlane(Vec() As Double, Pln() As Double, Prj() As Double)
    'project a vector onto a plane
    Dim D As Double, PlnIjk(2) As Double
    
    Call vNorm(Pln, PlnIjk)  'unit normal
    D = Vec(0) * PlnIjk(0) + Vec(1) * PlnIjk(1) + Vec(2) * PlnIjk(2)  'scalar product
    Prj(0) = Vec(0) - D * PlnIjk(0)
    Prj(1) = Vec(1) - D * PlnIjk(1)
    Prj(2) = Vec(2) - D * PlnIjk(2)
End Sub

Private Function Degrees(Ang As Double) As Double
    'converts radians to decimal degrees
    Degrees = Ang * 180 / PI
End Function

Private Function Radians(Ang As Double) As Double
    'converts decimal degrees to radians
    Radians = Ang * PI / 180
End Function

Private Sub RotVecAxis(V() As Double, N() As Double, DD As Double, U() As Double)
    'rotate a vector around an axis per Rodriques' rotation formula
    'V(2) = vector to rotate
    'N(2) = axis to rotate about
    'DD   = decimal degree angle to rotate CW about N
    'U(2) = vector result
    
    Dim Vp(2) As Double  'V projected onto N
    Dim Vr(2) As Double  'V projected onto plane at base of N
    Dim W(2) As Double   'vector perp to N & V
    Dim Vrr(2) As Double 'Vr vec rotated by A
    Dim dp As Double     'temp dot product
    Dim A As Double      'decimal degrees converted to radians
    
    Call vNorm(N, N)
    A = Radians(DD)
    
    'project V onto N axis using dot product (paral)
    dp = V(0) * N(0) + V(1) * N(1) + V(2) * N(2)
    Vp(0) = N(0) * dp
    Vp(1) = N(1) * dp
    Vp(2) = N(2) * dp
    
    'project V onto plane at base of N axis (perp)
    'this vector lies on base plane of N
    Vr(0) = V(0) - Vp(0)
    Vr(1) = V(1) - Vp(1)
    Vr(2) = V(2) - Vp(2)
    
    'create vector perpendicular to N and V
    'this cross product has same length as Vr
    'this vector lies on base plane of N
    Call vCross(N, V, W)
    
    'perform a "2D" rotation of Vr in the N base plane
    Vrr(0) = Vr(0) * Cos(A) + W(0) * Sin(A)
    Vrr(1) = Vr(1) * Cos(A) + W(1) * Sin(A)
    Vrr(2) = Vr(2) * Cos(A) + W(2) * Sin(A)
    
    'add Vrr to Vp
    U(0) = Vrr(0) + Vp(0)
    U(1) = Vrr(1) + Vp(1)
    U(2) = Vrr(2) + Vp(2)
End Sub

Private Sub vCross(A() As Double, B() As Double, C() As Double)
    ' cross product of a with b
    C(0) = A(1) * B(2) - A(2) * B(1)
    C(1) = A(2) * B(0) - A(0) * B(2)
    C(2) = A(0) * B(1) - A(1) * B(0)
End Sub

Private Function vDotAng(A() As Double, B() As Double) As Double
    ' return the angle (decimal degrees) between 2 vectors
    Dim U As Double, V As Double, W As Double
    Dim M As Double, N As Double
    'vector magnitudes
    U = Sqr(A(0) ^ 2 + A(1) ^ 2 + A(2) ^ 2)
    V = Sqr(B(0) ^ 2 + B(1) ^ 2 + B(2) ^ 2)
    'scalar product
    W = A(0) * B(0) + A(1) * B(1) + A(2) * B(2)
    'get angle
    vDotAng = Degrees(ArcCos(W / (U * V)))
End Function

Private Sub vNorm(A() As Double, B() As Double)
    ' normalize a vector to a unit length of 1
    Dim Mag As Double
    Mag = Sqr(A(0) ^ 2 + A(1) ^ 2 + A(2) ^ 2)
    B(0) = A(0) / Mag
    B(1) = A(1) / Mag
    B(2) = A(2) / Mag
End Sub

Reply

GomoFazter over 8 years ago

Code part 2 of 2

Private Sub GetProbeHeadMatrix(a0b0 As String, a90b180 As String, pMat() As Double)
    'determine 3x3 probe matrix given 2 initial probe rotation directions
    'creates frame to transform from head axis to machine axis
    'a0b0 & a90b180 represent directions pointed towards in PcDmis
    'allowed strings: XMINUS, XPLUS, YMINUS, YPLUS, ZMINUS, ZPLUS
    
    Dim Val As Integer, Axis As Integer
    
    'zero out rotation matrix
    pMat(0, 0) = 0: pMat(0, 1) = 0: pMat(0, 2) = 0
    pMat(2, 0) = 0: pMat(2, 1) = 0: pMat(2, 2) = 0
    
    'a0b0 for probe body Z axis
    Val = 1
    If UCase(Mid(a0b0, 2, 1)) = "P" Then Val = -Val  'invert direction
    Axis = Asc(UCase(Left$(a0b0, 1))) - 88
    pMat(2, Axis) = Val
    
    'a90b180 for probe body X axis
    Val = 1
    If UCase(Mid(a90b180, 2, 1)) = "P" Then Val = -Val  'invert direction
    Axis = Asc(UCase(Left$(a90b180, 1))) - 88
    pMat(0, Axis) = Val
    
    'cross product for probe body Y axis
    pMat(1, 0) = pMat(2, 1) * pMat(0, 2) - pMat(2, 2) * pMat(0, 1)
    pMat(1, 1) = pMat(2, 2) * pMat(0, 0) - pMat(2, 0) * pMat(0, 2)
    pMat(1, 2) = pMat(2, 0) * pMat(0, 1) - pMat(2, 1) * pMat(0, 0)
End Sub

Private Sub PullMatRow(M() As Double, Row As Integer, V() As Double)
    'extract a row from a matrix (dumb VBA)
    V(0) = M(Row, 0)
    V(1) = M(Row, 1)
    V(2) = M(Row, 2)
End Sub

Private Function ArcCos(X As Double) As Double
    'return the arc cosine
    ArcCos = 0
    If X < 1 Then
        If X > -1 Then
            ArcCos = Atn(-X / Sqr(-X * X + 1)) + 2 * Atn(1)
        Else
            ArcCos = 180
        End If
    End If
End Function

Private Sub MatMult(A() As Double, B() As Double, C() As Double)
    'matrix multiplication - takes 2 3x3 matrixes
    'a #columns = b #rows,  c = a #rows & b #columns
    C(0, 0) = A(0, 0) * B(0, 0) + A(0, 1) * B(1, 0) + A(0, 2) * B(2, 0)
    C(0, 1) = A(0, 0) * B(0, 1) + A(0, 1) * B(1, 1) + A(0, 2) * B(2, 1)
    C(0, 2) = A(0, 0) * B(0, 2) + A(0, 1) * B(1, 2) + A(0, 2) * B(2, 2)
    C(1, 0) = A(1, 0) * B(0, 0) + A(1, 1) * B(1, 0) + A(1, 2) * B(2, 0)
    C(1, 1) = A(1, 0) * B(0, 1) + A(1, 1) * B(1, 1) + A(1, 2) * B(2, 1)
    C(1, 2) = A(1, 0) * B(0, 2) + A(1, 1) * B(1, 2) + A(1, 2) * B(2, 2)
    C(2, 0) = A(2, 0) * B(0, 0) + A(2, 1) * B(1, 0) + A(2, 2) * B(2, 0)
    C(2, 1) = A(2, 0) * B(0, 1) + A(2, 1) * B(1, 1) + A(2, 2) * B(2, 1)
    C(2, 2) = A(2, 0) * B(0, 2) + A(2, 1) * B(1, 2) + A(2, 2) * B(2, 2)
End Sub

Private Sub ProjectVecOntoPlane(Vec() As Double, Pln() As Double, Prj() As Double)
    'project a vector onto a plane
    Dim D As Double, PlnIjk(2) As Double
    
    Call vNorm(Pln, PlnIjk)  'unit normal
    D = Vec(0) * PlnIjk(0) + Vec(1) * PlnIjk(1) + Vec(2) * PlnIjk(2)  'scalar product
    Prj(0) = Vec(0) - D * PlnIjk(0)
    Prj(1) = Vec(1) - D * PlnIjk(1)
    Prj(2) = Vec(2) - D * PlnIjk(2)
End Sub

Private Function Degrees(Ang As Double) As Double
    'converts radians to decimal degrees
    Degrees = Ang * 180 / PI
End Function

Private Function Radians(Ang As Double) As Double
    'converts decimal degrees to radians
    Radians = Ang * PI / 180
End Function

Private Sub RotVecAxis(V() As Double, N() As Double, DD As Double, U() As Double)
    'rotate a vector around an axis per Rodriques' rotation formula
    'V(2) = vector to rotate
    'N(2) = axis to rotate about
    'DD   = decimal degree angle to rotate CW about N
    'U(2) = vector result
    
    Dim Vp(2) As Double  'V projected onto N
    Dim Vr(2) As Double  'V projected onto plane at base of N
    Dim W(2) As Double   'vector perp to N & V
    Dim Vrr(2) As Double 'Vr vec rotated by A
    Dim dp As Double     'temp dot product
    Dim A As Double      'decimal degrees converted to radians
    
    Call vNorm(N, N)
    A = Radians(DD)
    
    'project V onto N axis using dot product (paral)
    dp = V(0) * N(0) + V(1) * N(1) + V(2) * N(2)
    Vp(0) = N(0) * dp
    Vp(1) = N(1) * dp
    Vp(2) = N(2) * dp
    
    'project V onto plane at base of N axis (perp)
    'this vector lies on base plane of N
    Vr(0) = V(0) - Vp(0)
    Vr(1) = V(1) - Vp(1)
    Vr(2) = V(2) - Vp(2)
    
    'create vector perpendicular to N and V
    'this cross product has same length as Vr
    'this vector lies on base plane of N
    Call vCross(N, V, W)
    
    'perform a "2D" rotation of Vr in the N base plane
    Vrr(0) = Vr(0) * Cos(A) + W(0) * Sin(A)
    Vrr(1) = Vr(1) * Cos(A) + W(1) * Sin(A)
    Vrr(2) = Vr(2) * Cos(A) + W(2) * Sin(A)
    
    'add Vrr to Vp
    U(0) = Vrr(0) + Vp(0)
    U(1) = Vrr(1) + Vp(1)
    U(2) = Vrr(2) + Vp(2)
End Sub

Private Sub vCross(A() As Double, B() As Double, C() As Double)
    ' cross product of a with b
    C(0) = A(1) * B(2) - A(2) * B(1)
    C(1) = A(2) * B(0) - A(0) * B(2)
    C(2) = A(0) * B(1) - A(1) * B(0)
End Sub

Private Function vDotAng(A() As Double, B() As Double) As Double
    ' return the angle (decimal degrees) between 2 vectors
    Dim U As Double, V As Double, W As Double
    Dim M As Double, N As Double
    'vector magnitudes
    U = Sqr(A(0) ^ 2 + A(1) ^ 2 + A(2) ^ 2)
    V = Sqr(B(0) ^ 2 + B(1) ^ 2 + B(2) ^ 2)
    'scalar product
    W = A(0) * B(0) + A(1) * B(1) + A(2) * B(2)
    'get angle
    vDotAng = Degrees(ArcCos(W / (U * V)))
End Function

Private Sub vNorm(A() As Double, B() As Double)
    ' normalize a vector to a unit length of 1
    Dim Mag As Double
    Mag = Sqr(A(0) ^ 2 + A(1) ^ 2 + A(2) ^ 2)
    B(0) = A(0) / Mag
    B(1) = A(1) / Mag
    B(2) = A(2) / Mag
End Sub

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