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.
#	Content
1	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