REPOS_ERICCA/magicsld/Outils_Math.vb

Module Outils_Math


#Region "calcul matriciel & vectoriel"

    Public Function Addition(ByRef A(,) As Double, ByRef B(,) As Double) As Double(,)
        'additionne deux matrices et le résultat est renvoyé dans la fonction

        ' vérification que l'on peut additionner, si toutes les matrices sont de 1 à nb, on devrait pas faire ça car ça bouffe du temps...
        Dim ua1 As Double, ub1 As Double
        Dim ua2 As Double, ub2 As Double

        ua1 = UBound(A, 1)
        ua2 = UBound(A, 2)
        ub1 = UBound(B, 1)
        ub2 = UBound(B, 2)

        If (Not ua1 - ub1 = 0) Or (Not ua2 - ub2 = 0) Then
            MsgBox("deux matrices ne peuvent pas s'additionner")
            Return Nothing
        End If

        Dim matC(ua1, ua2) As Double

        Dim i, j As Integer
        For i = 0 To ub1
            For j = 0 To ub2
                matC(i, j) = A(i, j) + B(i, j)
            Next j
        Next i

        Addition = matC
    End Function

    Public Function addition(ByRef a() As Double, ByRef b() As Double) As Double()
        ' addition de vecteurs

        If Not UBound(a) = UBound(b) Then
            MsgBox("Deux vecteurs ne peuvent s'additionner...")
            Return Nothing
        End If

        Dim i As Integer
        Dim c(UBound(a)) As Double

        For i = 0 To UBound(a)
            c(i) = a(i) + b(i)
        Next

        Return c
    End Function

    Public Function multiscal(ByRef A(,) As Double, ByRef c As Double) As Double(,)
        ' multiplie la matrice a par le scalaire c
        Dim i As Integer
        Dim j As Integer

        Dim B(UBound(A, 1), UBound(A, 2)) As Double

        For i = 0 To UBound(A, 1)
            For j = 0 To UBound(A, 2)
                B(i, j) = A(i, j) * c
            Next j
        Next i

        multiscal = B
    End Function

    Public Function multiscal(ByRef A() As Double, ByRef c As Double) As Double()
        'multiplie un vecteur par une certaine valeur
        Dim i As Integer
        Dim b(UBound(A)) As Double
        For i = 0 To UBound(A)
            b(i) = A(i) * c
        Next

        multiscal = b
    End Function

    Public Function reduit(ByRef A(,) As Double, Optional ByVal colonne As Integer = -1, Optional ByRef ligne As Integer = -1) As Double(,)
        ' fonction qui retire une colonne et /ou une ligne d'une matrice
        Dim temp(UBound(A, 1) - 1, UBound(A, 2) - 1) As Double
        Dim i As Integer, j As Integer, k As Integer, l As Integer


        For l = 0 To UBound(A)
            If Not (l = ligne) Then
                j = 0
                For k = 0 To UBound(A)
                    If Not (k = colonne) Then
                        temp(i, j) = A(l, k)
                        j += 1
                    End If
                Next k
                i += 1
            End If
        Next l


        reduit = temp
    End Function

    Public Function reduit(ByRef A() As Double, Optional ByVal colonne As Integer = -1) As Double()
        ' fonction qui retire un élément d'un vecteur
        Dim temp(UBound(A)) As Double
        Dim i As Double, l As Double

        While l < UBound(A)
            If Not l = colonne Then
                temp(i) = A(l)
                l = l + 1
                i = i + 1
            Else
                i = i + 1
            End If
        End While

        Return temp
    End Function

    Public Function multiplication(ByVal A(,) As Double, ByVal B(,) As Double) As Double(,)
        ' le vecteur C est redimensionné correctement
        ' [A]*[B] = [C]
        ' si B est un vecteur alors on va à la partie en bas

        Dim i, j, k, n, m As Integer
        Dim c(,) As Double

        m = UBound(B, 2)
        n = UBound(A, 1)

        If UBound(B, 1) <> UBound(A, 2) Then MsgBox(" Incompatibilité de matrices pour la multiplication") : Return Nothing

        ReDim c(n, m)

        For j = 0 To m
            For i = 0 To n
                For k = 0 To UBound(A, 2)
                    c(i, j) = c(i, j) + A(i, k) * B(k, j)
                Next k
            Next i
        Next j

        multiplication = c
    End Function

    Public Function multiplication(ByRef A(,) As Double, ByRef b() As Double) As Double()
        ' le vecteur C est redimensionné correctement
        ' [A]*{B} = {C}
        ' B est un vecteur ligne

        Dim i, j, n, m As Integer
        Dim c() As Double

        m = UBound(b)
        n = UBound(A, 1)

        If UBound(A, 2) <> m Then MsgBox("Impossible de multiplier la matrice avec le vecteur, dimensions incompatibles!") : Return Nothing

        ' on multiplie une matrice et un vecteur

        ReDim c(m)

        For i = 0 To UBound(b)
            For j = 0 To UBound(b)
                c(i) = c(i) + A(i, j) * b(j)
            Next j
        Next i
        Return c
    End Function

    Public Function multiplication(ByRef a() As Double, ByRef B(,) As Double) As Double()
        ' le vecteur C est redimensionné correctement
        ' {A}*[B] = {C}
        ' A est un vecteur  colonne

        Dim i, j, n, m As Integer
        Dim c() As Double

        m = UBound(a)
        n = UBound(B, 2)

        If m <> UBound(B, 1) Then MsgBox("Impossible de multiplier le veteur par la matrice.") : Return Nothing

        ReDim c(m)

        For i = 0 To UBound(B)
            For j = 0 To UBound(B)
                c(i) = c(i) + a(j) * B(j, i)
            Next j
        Next i
        Return c
    End Function

    Public Function soustraction(ByRef A(,) As Double, ByRef B(,) As Double) As Double(,)
        'C = A - B
        Dim C(,) As Double
        Dim m As Integer, n As Integer
        m = UBound(A, 1)
        n = UBound(A, 2)

        If (m <> UBound(B, 1)) Or (n <> UBound(B, 2)) Then MsgBox("Impossible de soustraire les matrices, dimensions non compatibles") : Return Nothing

        ReDim C(m, n)

        Dim i, j As Integer
        For i = 0 To m
            For j = 0 To n
                C(i, j) = A(i, j) - B(i, j)
            Next j
        Next i
        Return C
    End Function

    Public Function transpose(ByRef A(,) As Double) As Double(,)

        Dim c(,) As Double
        Dim i, j As Integer
        ReDim c(UBound(A, 2), UBound(A, 1))

        For i = 0 To UBound(A, 1)
            For j = 1 To UBound(A, 2)
                c(j, i) = A(i, j)
            Next j
        Next i

        Return c
    End Function

    Public Function gauss(ByRef A(,) As Double, ByRef C() As Double) As Double()
        ' [A]{X} = {C}
        ' les matrices doivent être déjà dimensionnées
        Dim n As Integer
        Dim i, j, k As Integer
        Dim rapport As Double

        n = UBound(A, 1)
        Dim rep() As Double
        ReDim rep(n)


        For i = 0 To n              ' de gauche à droite
            pivoter(i, A, C)

            'Call Form1.affiche(mat(), C())
            'MsgBox i

            For j = i + 1 To n      ' les lignes
                rapport = A(j, i) / A(i, i)
                For k = 1 To n      ' les colonnes
                    A(j, k) = A(j, k) - A(i, k) * rapport
                Next k
                C(j) = C(j) - C(i) * rapport
            Next j

        Next i


        Dim stuff As Double
        For i = n To 0 Step -1
            stuff = 0
            For j = i + 1 To n
                stuff = stuff + A(i, j) * rep(j)
            Next j
            rep(i) = (C(i) - stuff) / A(i, i)
        Next i
        C = rep
        gauss = rep
    End Function

    Private Sub pivoter(ByVal i As Integer, ByRef mat(,) As Double, ByRef C() As Double)
        ' sub qui pivote la ligne i de la matrice a ( et du vecteur C).  Si et seulement si le pivotage est nécessaire.
        Dim n As Integer
        n = UBound(mat, 1)
        Dim a_pivoter As Integer

        Dim z As Integer
        Dim temp As Double
        Dim tempC As Double

        For z = i To n
            temp = mat(i, i)

            If mat(z, i) > Math.Abs(temp) Then ' on a trouvé une ligneà pivoter
                temp = mat(z, i)
                a_pivoter = z
            End If
        Next z
        'MsgBox a_pivoter

        If a_pivoter <> 0 Then ' on pivote
            ' on pivote la ligne i et la ligne a_pivoter
            tempC = C(i)
            C(i) = C(a_pivoter)
            C(a_pivoter) = tempC

            For z = 0 To n
                temp = mat(i, z)
                mat(i, z) = mat(a_pivoter, z)
                mat(a_pivoter, z) = temp
            Next z

        End If
        ' fin pivot
    End Sub

    Private Function pivoter(ByVal i As Integer, ByRef mat(,) As Double) As Integer
        ' sub qui pivote la ligne i de la matrice a ( et du vecteur C).  Si et seulement si le pivotage est nécessaire.
        Dim n As Integer
        n = UBound(mat, 1)
        Dim a_pivoter As Integer
        pivoter = 1

        Dim z As Integer
        Dim temp As Double

        For z = i To n
            temp = mat(i, i)

            If mat(z, i) > Math.Abs(temp) Then ' on a trouvé une ligne à pivoter
                temp = mat(z, i)
                a_pivoter = z
                pivoter = -1
            End If
        Next z

        If a_pivoter <> 0 Then ' on pivote

            For z = 0 To n
                temp = mat(i, z)
                mat(i, z) = mat(a_pivoter, z)
                mat(a_pivoter, z) = temp
            Next z

        End If
        ' fin pivot
    End Function

    Public Function norme(ByRef u() As Double) As Double
        ' calcul la norme (longueur) d'un vecteur à xDimensions
        Dim i As Integer
        Dim temp As Double

        For i = 0 To UBound(u)
            temp = temp + u(i) ^ 2
        Next i
        Return Math.Sqrt(temp)
    End Function

    Public Function Prod_scalaire(ByRef u() As Double, ByRef v() As Double) As Double
        ' fait le produit scalaire entre deux vecteurs
        ' en notation indicielle: w = uivi
        Dim i As Integer

        For i = 0 To UBound(u)
            Prod_scalaire = Prod_scalaire + u(i) * v(i)
        Next i

    End Function

    Public Function prod_vect3D(ByRef u() As Double, ByRef v() As Double) As Double()
        ' fait le produit vectoriel entre les vecteur 3D u et v
        ' [ i  j  k  ]
        ' | u1 u2 u3 |
        ' [ v1 v2 v3 ]
        '
        ' w = (u2*v3 - u3*v2) i - (u1*v3 - u3*v1) j + (u1*v2-u2*v1) k
        Dim w(2) As Double
        w(0) = u(2) * v(3) - u(3) * v(2)
        w(1) = -(u(1) * v(3) - u(3) * v(1))
        w(2) = u(1) * v(2) - u(2) * v(1)

        prod_vect3D = w

    End Function

    Public Function unitaire(ByRef u() As Double) As Double()
        ' prend le vecteur u et son vecteur unitaire v
        Dim longueur As Double
        Dim i As Integer
        Dim v() As Double
        ReDim v(UBound(u))

        Dim temp As Double
        For i = 0 To UBound(u)
            temp = temp + u(i) ^ 2
        Next i
        longueur = Math.Sqrt(temp)
        For i = 0 To UBound(u)
            v(i) = u(i) / longueur
        Next i

        unitaire = v
    End Function

    Public Function cosdir(ByRef u() As Double, ByRef v() As Double) As Double
        ' retourne le cosinus entre deux vecteurs, u et v
        cosdir = Prod_scalaire(u, v) / norme(u) / norme(v)
    End Function

    Public Function projection(ByVal u() As Double, ByVal e() As Double) As Double
        ' projette le vecteur u sur e et retourne
        ' le résultat est la longueur sur le vecteur e
        projection = norme(u) * cosdir(u, e)
    End Function


    Public Function det(ByRef A(,) As Double) As Double
        ' calcule le détrminant de la matrice A
        Dim i As Long
        Dim temp As Double
        ' contrôle d'erreur:
        If UBound(A, 1) <> UBound(A, 2) Then
            MsgBox("Impossible de calculer le déterminant, la matrice n'est pas carrée")
        End If

        Dim n As Integer
        Dim j As Integer, k As Integer
        Dim rapport As Double

        n = UBound(A, 1)
        Dim rep() As Double
        ReDim rep(n)

        temp = 1
        For i = 0 To n              ' de gauche à droite
            temp *= pivoter(i, A)

            For j = i + 1 To n      ' les lignes
                rapport = A(j, i) / A(i, i)
                For k = 1 To n      ' les colonnes
                    A(j, k) = A(j, k) - A(i, k) * rapport
                Next k
            Next j
        Next i

        ' à partir d'ici on a une matrice triangulaire suppérieure. (mais je me casse pas la tête à mettre des 0...)

        For i = 0 To UBound(A)
            temp *= A(i, i)
        Next

        Return temp
    End Function


    Public Function det2(ByRef A(,) As Double) As Double
        ' calcul du déterminant à l'aide du cofacteur

        Dim i As Long
        Dim temp As Double
        ' contrôle d'erreur:
        If UBound(A, 1) <> UBound(A, 2) Then
            MsgBox("Impossible de calculer le déterminant, la matrice n'est pas carrée")
        End If

        If UBound(A) = 0 Then det2 = A(0, 0) : Exit Function
        Dim B(,) As Double
        ReDim B(UBound(A) - 1, UBound(A) - 1)


        For i = 0 To UBound(A)
            B = (reduit(A, i, 0))
            temp += (-1) ^ i * det2(reduit(A, i, 0)) * A(0, i)
        Next i

        det2 = temp

    End Function


    'Function qui compare 2 vecteurs.  retourne 0 si non alignés.  1 si dans le même sens, 2 si de sens opposé.
    Public Function CompareSens(ByRef u() As Double, ByRef v() As Double, Optional ByVal Epsilon As Double = 0.000000001) As Integer
        If ((u(0) - v(0)) < Epsilon) And ((u(1) - v(1)) < Epsilon) And ((u(2) - v(2)) < Epsilon) Then Return 1
        If ((u(0) + v(0)) < Epsilon) And ((u(1) + v(1)) < Epsilon) And ((u(2) + v(2)) < Epsilon) Then Return 2
        Return 0
    End Function

#End Region

#Region "calcul intégral"


    Function rectangles(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double
        ' function qui calcule l'intégrale de f par la méthode des rectangles
        Dim ecart As Double
        Dim i As Integer
        Dim nb As Long
        Dim x As Double
        Dim aire As Double

        ecart = b - a

        nb = CLng(ecart / dx)
        dx = ecart / nb

        If nb < 100 Then MsgBox("Attention, moins de 100 rectangles seront utilisés pour calculer l'intégrale")

        x = a
        For i = 1 To nb
            aire += f(x + dx / 2) * dx
            x += dx
        Next

        rectangles = aire

    End Function


    Function trapezes(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double
        ' function qui calcule l'intégrale de f par la méthode des trapezes
        Dim ecart As Double
        Dim i As Integer
        Dim nb As Long
        Dim x As Double
        Dim aire As Double

        ecart = b - a
        nb = CLng(ecart / dx)
        dx = ecart / nb

        If nb < 100 Then MsgBox("Attention, moins de 100 trapezes seront utilisés pour calculer l'intégrale")

        x = a
        For i = 1 To nb
            aire += (f(x) + f(x + dx)) / 2 * dx
            x += dx
        Next

        trapezes = aire
    End Function
    Function IntegrationGauss(ByRef n As Integer, Optional ByRef noFunction As Integer = 1) As Double
        Dim i As Integer
        Dim temp As Double
        Dim t() As Double
        Dim w() As Double
        ReDim w(0)
        ReDim t(0)

        Select Case n
            Case 1 ' un seul point d'intégration
                ReDim t(0)
                ReDim w(0)

                t(0) = 0
                w(0) = 2

            Case 2 ' deux points d'intégration
                ' somme wi f(ri)
                ReDim t(1)
                ReDim w(1)

                t(0) = -0.57735027
                t(1) = 0.57735027

                w(0) = 1
                w(1) = 1

            Case 3 ' trois points
                ReDim t(2)
                ReDim w(2)

                t(0) = -0.77459667
                t(1) = 0
                t(2) = 0.77459667

                w(0) = 5 / 9
                w(1) = 8 / 9
                w(2) = 5 / 9

            Case 4

                ReDim t(3)
                ReDim w(3)

                t(0) = -0.86113631
                t(1) = -0.33998104
                t(2) = 0.33998104
                t(3) = 0.86113631

                w(0) = 0.34785485
                w(1) = 0.65214515
                w(2) = 0.65214515
                w(3) = 0.34785485


            Case 5
                ReDim t(4)
                ReDim w(4)

                t(0) = -0.90617975
                t(1) = -0.53846931
                t(2) = 0
                t(3) = 0.53846931
                t(4) = 0.90617975

                w(0) = 0.23692689
                w(1) = 0.47862867
                w(2) = 0.56888889
                w(3) = 0.47862867
                w(4) = 0.23692689

            Case Else
                MsgBox("Problème dans l'intégration de Gauss, nb de points d'intégration non reconnu!?!")
        End Select


        For i = 0 To n - 1
            temp = temp + w(i) * Choose(noFunction, f1(t(i)), f2(t(i)), f3(t(i)), f4(t(i)), f5(t(i)))
        Next

        IntegrationGauss = temp
    End Function


    Function simpson(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double


    End Function

#End Region


    Public Function f(ByVal x As Double) As Double
        f = x ^ 2
    End Function
    Public Function f1(ByVal r As Double) As Double
        f1 = -0.04 * (-3 + r) * r ^ 2
    End Function
    Public Function f2(ByVal r As Double) As Double
        f2 = -0.04 * (-3 + r) * r ^ 2
    End Function
    Public Function f3(ByVal r As Double) As Double
        f3 = -0.04 * (-3 + r) * r ^ 2
    End Function
    Public Function f4(ByVal r As Double) As Double
        f4 = -0.04 * (-3 + r) * r ^ 2
    End Function
    Public Function f5(ByVal r As Double) As Double
        f5 = -0.04 * (-3 + r) * r ^ 2
    End Function


    ' Secant
    Public Function Sec(ByVal X As Double) As Double
        Sec = 1 / Math.Cos(X)
    End Function

    ' Cosecant
    Public Function CoSec(ByVal X As Double) As Double
        CoSec = 1 / Math.Sin(X)
    End Function

    ' Cotangent
    Public Function CoTan(ByVal X As Double) As Double
        CoTan = 1 / Math.Tan(X)
    End Function

    ' Inverse Sine
    Public Function ArcSin(ByVal X As Double) As Double
        ArcSin = Math.Atan(X / Math.Sqrt(-X * X + 1))
    End Function

    ' Inverse Cosine
    Public Function ArcCos(ByVal X As Double) As Double
        ArcCos = Math.Atan(-X / Math.Sqrt(-X * X + 1)) + 2 * Math.Atan(1)
    End Function

    ' Inverse Secant
    Public Function ArcSec(ByVal X As Double) As Double
        ArcSec = Math.Atan(X / Math.Sqrt(X * X - 1)) + Math.Sign(X - 1) * (2 * Math.Atan(1))
    End Function

    ' Inverse Cosecant
    Public Function ArcCoSec(ByVal X As Double) As Double
        ArcCoSec = Math.Atan(X / Math.Sqrt(X * X - 1)) + (Math.Sign(X) - 1) * (2 * Math.Atan(1))
    End Function

    ' Inverse Cotangent
    Public Function ArcCoTan(ByVal X As Double) As Double
        ArcCoTan = Math.Atan(X) + 2 * Math.Atan(1)
    End Function


    ' Hyperbolic Secant
    Public Function HSec(ByVal X As Double) As Double
        HSec = 2 / (Math.Exp(X) + Math.Exp(-X))
    End Function

    ' Hyperbolic Cosecant
    Public Function HCoSec(ByVal X As Double) As Double
        HCoSec = 2 / (Math.Exp(X) - Math.Exp(-X))
    End Function

    ' Hyperbolic Cotangent
    Public Function HCotan(ByVal X As Double) As Double
        HCotan = (Math.Exp(X) + Math.Exp(-X)) / (Math.Exp(X) - Math.Exp(-X))
    End Function

    ' Inverse Hyperbolic Sine
    Public Function HArcSin(ByVal X As Double) As Double
        HArcSin = Math.Log(X + Math.Sqrt(X * X + 1))
    End Function

    ' Inverse Hyperbolic Cosine
    Public Function HArcCos(ByVal X As Double) As Double
        HArcCos = Math.Log(X + Math.Sqrt(X * X - 1))
    End Function

    ' Inverse Hyperbolic Tangent
    Function HArcTan(ByVal X As Double) As Double
        HArcTan = Math.Log((1 + X) / (1 - X)) / 2
    End Function

    ' Inverse Hyperbolic Secant
    Public Function HArcSec(ByVal X As Double) As Double
        HArcSec = Math.Log((Math.Sqrt(-X * X + 1) + 1) / X)
    End Function

    ' Inverse Hyperbolic Cosecant
    Public Function HArcCoSec(ByVal X As Double) As Double
        HArcCoSec = Math.Log((Math.Sign(X) * Math.Sqrt(X * X + 1) + 1) / X)
    End Function

    ' Inverse Hyperbolic Cotangent
    Public Function HArcCoTan(ByVal X As Double) As Double
        HArcCoTan = Math.Log((X + 1) / (X - 1)) / 2
    End Function

    ' Logarithm to base N
    Public Function LogN(ByVal X As Double, ByVal n As Double) As Double
        LogN = Math.Log(X) / Math.Log(n)
    End Function

    ' function qui effectue la rotation d'un point Pt par la matrice Mat
    Public Function Rotation2D(ByRef Mat(,) As Double, ByVal Pt() As Double) As Double()
        Dim temp(1) As Double

        temp(0) = Mat(0, 0) * Pt(0) + Mat(0, 1) * Pt(1)
        temp(1) = Mat(0, 1) * Pt(0) + Mat(1, 1) * Pt(1)

        Return temp

    End Function

    ' function qui effectue la rotation d'un point Pt par le vecteur UNITAIRE u
    Public Function Rotation2D(ByRef u() As Double, ByVal Pt() As Double) As Double()
        Dim temp(1) As Double
        Dim longueur As Double = Math.Sqrt(u(0) * u(0) + u(1) * u(1))
        u(0) /= longueur
        u(1) /= longueur

        temp(0) = u(0) * Pt(0) - u(1) * Pt(1)
        temp(1) = u(1) * Pt(0) + u(0) * Pt(1)


        Return temp

    End Function

    Public Sub Rotation2D(ByRef u() As Double, ByRef Ptx As Double, ByRef Pty As Double)
        Dim temp(1) As Double
        Dim longueur As Double = Math.Sqrt(u(0) * u(0) + u(1) * u(1))
        u(0) /= longueur
        u(1) /= longueur

        temp(0) = u(0) * Ptx - u(1) * Pty
        temp(1) = u(1) * Ptx + u(0) * Pty

        Ptx = temp(0)
        Pty = temp(1)

    End Sub

    Public Function Angle2Vecteurs(ByRef u() As Double, ByRef v() As Double) As Double
        Dim temp As Double
        temp = Outils_Math.Prod_scalaire(u, v) / norme(u) / norme(v)
        If Math.Abs(temp - 1) < 0.000000000001 Or Math.Abs(temp + 1) < 0.000000000001 Then
            Return Pi
        Else
            Return Math.Acos(temp)
        End If
    End Function

End Module
Revision:	40
Committed:	Mon Aug 20 21:30:28 2007 UTC (17 years, 8 months ago) by bournival
File size:	23141 byte(s)
Log Message:	Projet de these de Sylvain Bournival. Attention projet VB.
#	User	Rev	Content
1	bournival	40	Module Outils_Math
2
3
4
5
6			#Region "calcul matriciel & vectoriel"
7
8			Public Function Addition(ByRef A(,) As Double, ByRef B(,) As Double) As Double(,)
9			'additionne deux matrices et le résultat est renvoyé dans la fonction
10
11			' vérification que l'on peut additionner, si toutes les matrices sont de 1 à nb, on devrait pas faire ça car ça bouffe du temps...
12			Dim ua1 As Double, ub1 As Double
13			Dim ua2 As Double, ub2 As Double
14
15			ua1 = UBound(A, 1)
16			ua2 = UBound(A, 2)
17			ub1 = UBound(B, 1)
18			ub2 = UBound(B, 2)
19
20			If (Not ua1 - ub1 = 0) Or (Not ua2 - ub2 = 0) Then
21			MsgBox("deux matrices ne peuvent pas s'additionner")
22			Return Nothing
23			End If
24
25			Dim matC(ua1, ua2) As Double
26
27			Dim i, j As Integer
28			For i = 0 To ub1
29			For j = 0 To ub2
30			matC(i, j) = A(i, j) + B(i, j)
31			Next j
32			Next i
33
34			Addition = matC
35			End Function
36
37			Public Function addition(ByRef a() As Double, ByRef b() As Double) As Double()
38			' addition de vecteurs
39
40			If Not UBound(a) = UBound(b) Then
41			MsgBox("Deux vecteurs ne peuvent s'additionner...")
42			Return Nothing
43			End If
44
45			Dim i As Integer
46			Dim c(UBound(a)) As Double
47
48			For i = 0 To UBound(a)
49			c(i) = a(i) + b(i)
50			Next
51
52			Return c
53			End Function
54
55			Public Function multiscal(ByRef A(,) As Double, ByRef c As Double) As Double(,)
56			' multiplie la matrice a par le scalaire c
57			Dim i As Integer
58			Dim j As Integer
59
60			Dim B(UBound(A, 1), UBound(A, 2)) As Double
61
62			For i = 0 To UBound(A, 1)
63			For j = 0 To UBound(A, 2)
64			B(i, j) = A(i, j) * c
65			Next j
66			Next i
67
68			multiscal = B
69			End Function
70
71			Public Function multiscal(ByRef A() As Double, ByRef c As Double) As Double()
72			'multiplie un vecteur par une certaine valeur
73			Dim i As Integer
74			Dim b(UBound(A)) As Double
75			For i = 0 To UBound(A)
76			b(i) = A(i) * c
77			Next
78
79			multiscal = b
80			End Function
81
82			Public Function reduit(ByRef A(,) As Double, Optional ByVal colonne As Integer = -1, Optional ByRef ligne As Integer = -1) As Double(,)
83			' fonction qui retire une colonne et /ou une ligne d'une matrice
84			Dim temp(UBound(A, 1) - 1, UBound(A, 2) - 1) As Double
85			Dim i As Integer, j As Integer, k As Integer, l As Integer
86
87
88			For l = 0 To UBound(A)
89			If Not (l = ligne) Then
90			j = 0
91			For k = 0 To UBound(A)
92			If Not (k = colonne) Then
93			temp(i, j) = A(l, k)
94			j += 1
95			End If
96			Next k
97			i += 1
98			End If
99			Next l
100
101
102			reduit = temp
103			End Function
104
105			Public Function reduit(ByRef A() As Double, Optional ByVal colonne As Integer = -1) As Double()
106			' fonction qui retire un élément d'un vecteur
107			Dim temp(UBound(A)) As Double
108			Dim i As Double, l As Double
109
110			While l < UBound(A)
111			If Not l = colonne Then
112			temp(i) = A(l)
113			l = l + 1
114			i = i + 1
115			Else
116			i = i + 1
117			End If
118			End While
119
120			Return temp
121			End Function
122
123			Public Function multiplication(ByVal A(,) As Double, ByVal B(,) As Double) As Double(,)
124			' le vecteur C est redimensionné correctement
125			' [A]*[B] = [C]
126			' si B est un vecteur alors on va à la partie en bas
127
128			Dim i, j, k, n, m As Integer
129			Dim c(,) As Double
130
131			m = UBound(B, 2)
132			n = UBound(A, 1)
133
134			If UBound(B, 1) <> UBound(A, 2) Then MsgBox(" Incompatibilité de matrices pour la multiplication") : Return Nothing
135
136			ReDim c(n, m)
137
138			For j = 0 To m
139			For i = 0 To n
140			For k = 0 To UBound(A, 2)
141			c(i, j) = c(i, j) + A(i, k) * B(k, j)
142			Next k
143			Next i
144			Next j
145
146			multiplication = c
147			End Function
148
149			Public Function multiplication(ByRef A(,) As Double, ByRef b() As Double) As Double()
150			' le vecteur C est redimensionné correctement
151			' [A]*{B} = {C}
152			' B est un vecteur ligne
153
154			Dim i, j, n, m As Integer
155			Dim c() As Double
156
157			m = UBound(b)
158			n = UBound(A, 1)
159
160			If UBound(A, 2) <> m Then MsgBox("Impossible de multiplier la matrice avec le vecteur, dimensions incompatibles!") : Return Nothing
161
162			' on multiplie une matrice et un vecteur
163
164			ReDim c(m)
165
166			For i = 0 To UBound(b)
167			For j = 0 To UBound(b)
168			c(i) = c(i) + A(i, j) * b(j)
169			Next j
170			Next i
171			Return c
172			End Function
173
174			Public Function multiplication(ByRef a() As Double, ByRef B(,) As Double) As Double()
175			' le vecteur C est redimensionné correctement
176			' {A}*[B] = {C}
177			' A est un vecteur colonne
178
179			Dim i, j, n, m As Integer
180			Dim c() As Double
181
182			m = UBound(a)
183			n = UBound(B, 2)
184
185			If m <> UBound(B, 1) Then MsgBox("Impossible de multiplier le veteur par la matrice.") : Return Nothing
186
187			ReDim c(m)
188
189			For i = 0 To UBound(B)
190			For j = 0 To UBound(B)
191			c(i) = c(i) + a(j) * B(j, i)
192			Next j
193			Next i
194			Return c
195			End Function
196
197			Public Function soustraction(ByRef A(,) As Double, ByRef B(,) As Double) As Double(,)
198			'C = A - B
199			Dim C(,) As Double
200			Dim m As Integer, n As Integer
201			m = UBound(A, 1)
202			n = UBound(A, 2)
203
204			If (m <> UBound(B, 1)) Or (n <> UBound(B, 2)) Then MsgBox("Impossible de soustraire les matrices, dimensions non compatibles") : Return Nothing
205
206			ReDim C(m, n)
207
208			Dim i, j As Integer
209			For i = 0 To m
210			For j = 0 To n
211			C(i, j) = A(i, j) - B(i, j)
212			Next j
213			Next i
214			Return C
215			End Function
216
217			Public Function transpose(ByRef A(,) As Double) As Double(,)
218
219			Dim c(,) As Double
220			Dim i, j As Integer
221			ReDim c(UBound(A, 2), UBound(A, 1))
222
223			For i = 0 To UBound(A, 1)
224			For j = 1 To UBound(A, 2)
225			c(j, i) = A(i, j)
226			Next j
227			Next i
228
229			Return c
230			End Function
231
232			Public Function gauss(ByRef A(,) As Double, ByRef C() As Double) As Double()
233			' [A]{X} = {C}
234			' les matrices doivent être déjà dimensionnées
235			Dim n As Integer
236			Dim i, j, k As Integer
237			Dim rapport As Double
238
239			n = UBound(A, 1)
240			Dim rep() As Double
241			ReDim rep(n)
242
243
244			For i = 0 To n ' de gauche à droite
245			pivoter(i, A, C)
246
247			'Call Form1.affiche(mat(), C())
248			'MsgBox i
249
250			For j = i + 1 To n ' les lignes
251			rapport = A(j, i) / A(i, i)
252			For k = 1 To n ' les colonnes
253			A(j, k) = A(j, k) - A(i, k) * rapport
254			Next k
255			C(j) = C(j) - C(i) * rapport
256			Next j
257
258			Next i
259
260
261			Dim stuff As Double
262			For i = n To 0 Step -1
263			stuff = 0
264			For j = i + 1 To n
265			stuff = stuff + A(i, j) * rep(j)
266			Next j
267			rep(i) = (C(i) - stuff) / A(i, i)
268			Next i
269			C = rep
270			gauss = rep
271			End Function
272
273			Private Sub pivoter(ByVal i As Integer, ByRef mat(,) As Double, ByRef C() As Double)
274			' sub qui pivote la ligne i de la matrice a ( et du vecteur C). Si et seulement si le pivotage est nécessaire.
275			Dim n As Integer
276			n = UBound(mat, 1)
277			Dim a_pivoter As Integer
278
279			Dim z As Integer
280			Dim temp As Double
281			Dim tempC As Double
282
283			For z = i To n
284			temp = mat(i, i)
285
286			If mat(z, i) > Math.Abs(temp) Then ' on a trouvé une ligneà pivoter
287			temp = mat(z, i)
288			a_pivoter = z
289			End If
290			Next z
291			'MsgBox a_pivoter
292
293			If a_pivoter <> 0 Then ' on pivote
294			' on pivote la ligne i et la ligne a_pivoter
295			tempC = C(i)
296			C(i) = C(a_pivoter)
297			C(a_pivoter) = tempC
298
299			For z = 0 To n
300			temp = mat(i, z)
301			mat(i, z) = mat(a_pivoter, z)
302			mat(a_pivoter, z) = temp
303			Next z
304
305			End If
306			' fin pivot
307			End Sub
308
309			Private Function pivoter(ByVal i As Integer, ByRef mat(,) As Double) As Integer
310			' sub qui pivote la ligne i de la matrice a ( et du vecteur C). Si et seulement si le pivotage est nécessaire.
311			Dim n As Integer
312			n = UBound(mat, 1)
313			Dim a_pivoter As Integer
314			pivoter = 1
315
316			Dim z As Integer
317			Dim temp As Double
318
319			For z = i To n
320			temp = mat(i, i)
321
322			If mat(z, i) > Math.Abs(temp) Then ' on a trouvé une ligne à pivoter
323			temp = mat(z, i)
324			a_pivoter = z
325			pivoter = -1
326			End If
327			Next z
328
329			If a_pivoter <> 0 Then ' on pivote
330
331			For z = 0 To n
332			temp = mat(i, z)
333			mat(i, z) = mat(a_pivoter, z)
334			mat(a_pivoter, z) = temp
335			Next z
336
337			End If
338			' fin pivot
339			End Function
340
341			Public Function norme(ByRef u() As Double) As Double
342			' calcul la norme (longueur) d'un vecteur à xDimensions
343			Dim i As Integer
344			Dim temp As Double
345
346			For i = 0 To UBound(u)
347			temp = temp + u(i) ^ 2
348			Next i
349			Return Math.Sqrt(temp)
350			End Function
351
352			Public Function Prod_scalaire(ByRef u() As Double, ByRef v() As Double) As Double
353			' fait le produit scalaire entre deux vecteurs
354			' en notation indicielle: w = uivi
355			Dim i As Integer
356
357			For i = 0 To UBound(u)
358			Prod_scalaire = Prod_scalaire + u(i) * v(i)
359			Next i
360
361			End Function
362
363			Public Function prod_vect3D(ByRef u() As Double, ByRef v() As Double) As Double()
364			' fait le produit vectoriel entre les vecteur 3D u et v
365			' [ i j k ]
366			' \| u1 u2 u3 \|
367			' [ v1 v2 v3 ]
368			'
369			' w = (u2v3 - u3v2) i - (u1v3 - u3v1) j + (u1v2-u2v1) k
370			Dim w(2) As Double
371			w(0) = u(2) * v(3) - u(3) * v(2)
372			w(1) = -(u(1) * v(3) - u(3) * v(1))
373			w(2) = u(1) * v(2) - u(2) * v(1)
374
375			prod_vect3D = w
376
377			End Function
378
379			Public Function unitaire(ByRef u() As Double) As Double()
380			' prend le vecteur u et son vecteur unitaire v
381			Dim longueur As Double
382			Dim i As Integer
383			Dim v() As Double
384			ReDim v(UBound(u))
385
386			Dim temp As Double
387			For i = 0 To UBound(u)
388			temp = temp + u(i) ^ 2
389			Next i
390			longueur = Math.Sqrt(temp)
391			For i = 0 To UBound(u)
392			v(i) = u(i) / longueur
393			Next i
394
395			unitaire = v
396			End Function
397
398			Public Function cosdir(ByRef u() As Double, ByRef v() As Double) As Double
399			' retourne le cosinus entre deux vecteurs, u et v
400			cosdir = Prod_scalaire(u, v) / norme(u) / norme(v)
401			End Function
402
403			Public Function projection(ByVal u() As Double, ByVal e() As Double) As Double
404			' projette le vecteur u sur e et retourne
405			' le résultat est la longueur sur le vecteur e
406			projection = norme(u) * cosdir(u, e)
407			End Function
408
409
410
411			Public Function det(ByRef A(,) As Double) As Double
412			' calcule le détrminant de la matrice A
413			Dim i As Long
414			Dim temp As Double
415			' contrôle d'erreur:
416			If UBound(A, 1) <> UBound(A, 2) Then
417			MsgBox("Impossible de calculer le déterminant, la matrice n'est pas carrée")
418			End If
419
420			Dim n As Integer
421			Dim j As Integer, k As Integer
422			Dim rapport As Double
423
424			n = UBound(A, 1)
425			Dim rep() As Double
426			ReDim rep(n)
427
428			temp = 1
429			For i = 0 To n ' de gauche à droite
430			temp *= pivoter(i, A)
431
432			For j = i + 1 To n ' les lignes
433			rapport = A(j, i) / A(i, i)
434			For k = 1 To n ' les colonnes
435			A(j, k) = A(j, k) - A(i, k) * rapport
436			Next k
437			Next j
438			Next i
439
440			' à partir d'ici on a une matrice triangulaire suppérieure. (mais je me casse pas la tête à mettre des 0...)
441
442			For i = 0 To UBound(A)
443			temp *= A(i, i)
444			Next
445
446			Return temp
447			End Function
448
449
450			Public Function det2(ByRef A(,) As Double) As Double
451			' calcul du déterminant à l'aide du cofacteur
452
453			Dim i As Long
454			Dim temp As Double
455			' contrôle d'erreur:
456			If UBound(A, 1) <> UBound(A, 2) Then
457			MsgBox("Impossible de calculer le déterminant, la matrice n'est pas carrée")
458			End If
459
460			If UBound(A) = 0 Then det2 = A(0, 0) : Exit Function
461			Dim B(,) As Double
462			ReDim B(UBound(A) - 1, UBound(A) - 1)
463
464
465			For i = 0 To UBound(A)
466			B = (reduit(A, i, 0))
467			temp += (-1) ^ i * det2(reduit(A, i, 0)) * A(0, i)
468			Next i
469
470			det2 = temp
471
472			End Function
473
474
475			'Function qui compare 2 vecteurs. retourne 0 si non alignés. 1 si dans le même sens, 2 si de sens opposé.
476			Public Function CompareSens(ByRef u() As Double, ByRef v() As Double, Optional ByVal Epsilon As Double = 0.000000001) As Integer
477			If ((u(0) - v(0)) < Epsilon) And ((u(1) - v(1)) < Epsilon) And ((u(2) - v(2)) < Epsilon) Then Return 1
478			If ((u(0) + v(0)) < Epsilon) And ((u(1) + v(1)) < Epsilon) And ((u(2) + v(2)) < Epsilon) Then Return 2
479			Return 0
480			End Function
481
482			#End Region
483
484			#Region "calcul intégral"
485
486
487			Function rectangles(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double
488			' function qui calcule l'intégrale de f par la méthode des rectangles
489			Dim ecart As Double
490			Dim i As Integer
491			Dim nb As Long
492			Dim x As Double
493			Dim aire As Double
494
495			ecart = b - a
496
497			nb = CLng(ecart / dx)
498			dx = ecart / nb
499
500			If nb < 100 Then MsgBox("Attention, moins de 100 rectangles seront utilisés pour calculer l'intégrale")
501
502			x = a
503			For i = 1 To nb
504			aire += f(x + dx / 2) * dx
505			x += dx
506			Next
507
508			rectangles = aire
509
510			End Function
511
512
513			Function trapezes(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double
514			' function qui calcule l'intégrale de f par la méthode des trapezes
515			Dim ecart As Double
516			Dim i As Integer
517			Dim nb As Long
518			Dim x As Double
519			Dim aire As Double
520
521			ecart = b - a
522			nb = CLng(ecart / dx)
523			dx = ecart / nb
524
525			If nb < 100 Then MsgBox("Attention, moins de 100 trapezes seront utilisés pour calculer l'intégrale")
526
527			x = a
528			For i = 1 To nb
529			aire += (f(x) + f(x + dx)) / 2 * dx
530			x += dx
531			Next
532
533			trapezes = aire
534			End Function
535			Function IntegrationGauss(ByRef n As Integer, Optional ByRef noFunction As Integer = 1) As Double
536			Dim i As Integer
537			Dim temp As Double
538			Dim t() As Double
539			Dim w() As Double
540			ReDim w(0)
541			ReDim t(0)
542
543			Select Case n
544			Case 1 ' un seul point d'intégration
545			ReDim t(0)
546			ReDim w(0)
547
548			t(0) = 0
549			w(0) = 2
550
551			Case 2 ' deux points d'intégration
552			' somme wi f(ri)
553			ReDim t(1)
554			ReDim w(1)
555
556			t(0) = -0.57735027
557			t(1) = 0.57735027
558
559			w(0) = 1
560			w(1) = 1
561
562			Case 3 ' trois points
563			ReDim t(2)
564			ReDim w(2)
565
566			t(0) = -0.77459667
567			t(1) = 0
568			t(2) = 0.77459667
569
570			w(0) = 5 / 9
571			w(1) = 8 / 9
572			w(2) = 5 / 9
573
574			Case 4
575
576			ReDim t(3)
577			ReDim w(3)
578
579			t(0) = -0.86113631
580			t(1) = -0.33998104
581			t(2) = 0.33998104
582			t(3) = 0.86113631
583
584			w(0) = 0.34785485
585			w(1) = 0.65214515
586			w(2) = 0.65214515
587			w(3) = 0.34785485
588
589
590			Case 5
591			ReDim t(4)
592			ReDim w(4)
593
594			t(0) = -0.90617975
595			t(1) = -0.53846931
596			t(2) = 0
597			t(3) = 0.53846931
598			t(4) = 0.90617975
599
600			w(0) = 0.23692689
601			w(1) = 0.47862867
602			w(2) = 0.56888889
603			w(3) = 0.47862867
604			w(4) = 0.23692689
605
606			Case Else
607			MsgBox("Problème dans l'intégration de Gauss, nb de points d'intégration non reconnu!?!")
608			End Select
609
610
611			For i = 0 To n - 1
612			temp = temp + w(i) * Choose(noFunction, f1(t(i)), f2(t(i)), f3(t(i)), f4(t(i)), f5(t(i)))
613			Next
614
615			IntegrationGauss = temp
616			End Function
617
618
619			Function simpson(ByRef dx As Double, Optional ByVal a As Double = 0, Optional ByVal b As Double = 1) As Double
620
621
622			End Function
623
624			#End Region
625
626
627
628
629			Public Function f(ByVal x As Double) As Double
630			f = x ^ 2
631			End Function
632			Public Function f1(ByVal r As Double) As Double
633			f1 = -0.04 * (-3 + r) * r ^ 2
634			End Function
635			Public Function f2(ByVal r As Double) As Double
636			f2 = -0.04 * (-3 + r) * r ^ 2
637			End Function
638			Public Function f3(ByVal r As Double) As Double
639			f3 = -0.04 * (-3 + r) * r ^ 2
640			End Function
641			Public Function f4(ByVal r As Double) As Double
642			f4 = -0.04 * (-3 + r) * r ^ 2
643			End Function
644			Public Function f5(ByVal r As Double) As Double
645			f5 = -0.04 * (-3 + r) * r ^ 2
646			End Function
647
648
649			' Secant
650			Public Function Sec(ByVal X As Double) As Double
651			Sec = 1 / Math.Cos(X)
652			End Function
653
654			' Cosecant
655			Public Function CoSec(ByVal X As Double) As Double
656			CoSec = 1 / Math.Sin(X)
657			End Function
658
659			' Cotangent
660			Public Function CoTan(ByVal X As Double) As Double
661			CoTan = 1 / Math.Tan(X)
662			End Function
663
664			' Inverse Sine
665			Public Function ArcSin(ByVal X As Double) As Double
666			ArcSin = Math.Atan(X / Math.Sqrt(-X * X + 1))
667			End Function
668
669			' Inverse Cosine
670			Public Function ArcCos(ByVal X As Double) As Double
671			ArcCos = Math.Atan(-X / Math.Sqrt(-X * X + 1)) + 2 * Math.Atan(1)
672			End Function
673
674			' Inverse Secant
675			Public Function ArcSec(ByVal X As Double) As Double
676			ArcSec = Math.Atan(X / Math.Sqrt(X * X - 1)) + Math.Sign(X - 1) * (2 * Math.Atan(1))
677			End Function
678
679			' Inverse Cosecant
680			Public Function ArcCoSec(ByVal X As Double) As Double
681			ArcCoSec = Math.Atan(X / Math.Sqrt(X * X - 1)) + (Math.Sign(X) - 1) * (2 * Math.Atan(1))
682			End Function
683
684			' Inverse Cotangent
685			Public Function ArcCoTan(ByVal X As Double) As Double
686			ArcCoTan = Math.Atan(X) + 2 * Math.Atan(1)
687			End Function
688
689
690			' Hyperbolic Secant
691			Public Function HSec(ByVal X As Double) As Double
692			HSec = 2 / (Math.Exp(X) + Math.Exp(-X))
693			End Function
694
695			' Hyperbolic Cosecant
696			Public Function HCoSec(ByVal X As Double) As Double
697			HCoSec = 2 / (Math.Exp(X) - Math.Exp(-X))
698			End Function
699
700			' Hyperbolic Cotangent
701			Public Function HCotan(ByVal X As Double) As Double
702			HCotan = (Math.Exp(X) + Math.Exp(-X)) / (Math.Exp(X) - Math.Exp(-X))
703			End Function
704
705			' Inverse Hyperbolic Sine
706			Public Function HArcSin(ByVal X As Double) As Double
707			HArcSin = Math.Log(X + Math.Sqrt(X * X + 1))
708			End Function
709
710			' Inverse Hyperbolic Cosine
711			Public Function HArcCos(ByVal X As Double) As Double
712			HArcCos = Math.Log(X + Math.Sqrt(X * X - 1))
713			End Function
714
715			' Inverse Hyperbolic Tangent
716			Function HArcTan(ByVal X As Double) As Double
717			HArcTan = Math.Log((1 + X) / (1 - X)) / 2
718			End Function
719
720			' Inverse Hyperbolic Secant
721			Public Function HArcSec(ByVal X As Double) As Double
722			HArcSec = Math.Log((Math.Sqrt(-X * X + 1) + 1) / X)
723			End Function
724
725			' Inverse Hyperbolic Cosecant
726			Public Function HArcCoSec(ByVal X As Double) As Double
727			HArcCoSec = Math.Log((Math.Sign(X) * Math.Sqrt(X * X + 1) + 1) / X)
728			End Function
729
730			' Inverse Hyperbolic Cotangent
731			Public Function HArcCoTan(ByVal X As Double) As Double
732			HArcCoTan = Math.Log((X + 1) / (X - 1)) / 2
733			End Function
734
735			' Logarithm to base N
736			Public Function LogN(ByVal X As Double, ByVal n As Double) As Double
737			LogN = Math.Log(X) / Math.Log(n)
738			End Function
739
740			' function qui effectue la rotation d'un point Pt par la matrice Mat
741			Public Function Rotation2D(ByRef Mat(,) As Double, ByVal Pt() As Double) As Double()
742			Dim temp(1) As Double
743
744			temp(0) = Mat(0, 0) * Pt(0) + Mat(0, 1) * Pt(1)
745			temp(1) = Mat(0, 1) * Pt(0) + Mat(1, 1) * Pt(1)
746
747			Return temp
748
749			End Function
750
751			' function qui effectue la rotation d'un point Pt par le vecteur UNITAIRE u
752			Public Function Rotation2D(ByRef u() As Double, ByVal Pt() As Double) As Double()
753			Dim temp(1) As Double
754			Dim longueur As Double = Math.Sqrt(u(0) * u(0) + u(1) * u(1))
755			u(0) /= longueur
756			u(1) /= longueur
757
758			temp(0) = u(0) * Pt(0) - u(1) * Pt(1)
759			temp(1) = u(1) * Pt(0) + u(0) * Pt(1)
760
761
762
763			Return temp
764
765			End Function
766
767			Public Sub Rotation2D(ByRef u() As Double, ByRef Ptx As Double, ByRef Pty As Double)
768			Dim temp(1) As Double
769			Dim longueur As Double = Math.Sqrt(u(0) * u(0) + u(1) * u(1))
770			u(0) /= longueur
771			u(1) /= longueur
772
773			temp(0) = u(0) * Ptx - u(1) * Pty
774			temp(1) = u(1) * Ptx + u(0) * Pty
775
776			Ptx = temp(0)
777			Pty = temp(1)
778
779			End Sub
780
781			Public Function Angle2Vecteurs(ByRef u() As Double, ByRef v() As Double) As Double
782			Dim temp As Double
783			temp = Outils_Math.Prod_scalaire(u, v) / norme(u) / norme(v)
784			If Math.Abs(temp - 1) < 0.000000000001 Or Math.Abs(temp + 1) < 0.000000000001 Then
785			Return Pi
786			Else
787			Return Math.Acos(temp)
788			End If
789			End Function
790
791			End Module