Adapt smoothing with splines to ODF1.2

2 files changed, 605 insertions, 158 deletions

diff --git a/chart2/source/view/charttypes/Splines.cxx b/chart2/source/view/charttypes/Splines.cxx
index 6b1d1007b700..56e7740f2d68 100644
--- a/chart2/source/view/charttypes/Splines.cxx
+++ b/chart2/source/view/charttypes/Splines.cxx
@@ -46,17 +46,6 @@ using namespace ::com::sun::star;
 namespace
 {
 
-template< typename T >
-struct lcl_EqualsFirstDoubleOfPair : ::std::binary_function< ::std::pair< double, T >, ::std::pair< double, T >, bool >
-{
-    inline bool operator() ( const ::std::pair< double, T > & rOne, const ::std::pair< double, T > & rOther )
-    {
-        return ( ::rtl::math::approxEqual( rOne.first, rOther.first ) );
-    }
-};
-
-//-----------------------------------------------------------------------------
-
 typedef ::std::pair< double, double >   tPointType;
 typedef ::std::vector< tPointType >     tPointVecType;
 typedef tPointVecType::size_type        lcl_tSizeType;
@@ -79,6 +68,16 @@ public:
                            double fY1FirstDerivation,
                            double fYnFirstDerivation );
 
+
+    /** @descr creates an object that calculates cublic splines on construction
+               for the special case of periodic cubic spline
+
+        @param rSortedPoints  the points for which splines shall be calculated,
+               they need to be sorted in x values. First and last y value must be equal
+     */
+    lcl_SplineCalculation( const tPointVecType & rSortedPoints);
+
+
     /** @descr this function corresponds to the function splint in [1].
 
         [1] Numerical Recipies in C, 2nd edition
@@ -98,8 +97,8 @@ private:
     double m_fYpN;
 
     // these values are cached for performance reasons
-    tPointVecType::size_type m_nKLow;
-    tPointVecType::size_type m_nKHigh;
+    lcl_tSizeType m_nKLow;
+    lcl_tSizeType m_nKHigh;
     double m_fLastInterpolatedValue;
 
     /** @descr this function corresponds to the function spline in [1].
@@ -109,6 +108,22 @@ private:
             Section 3.3, page 115
     */
     void Calculate();
+
+    /** @descr this function corresponds to the algoritm 4.76 in [2] and
+        theorem 5.3.7 in [3]
+
+        [2] Engeln-Müllges, Gisela: Numerik-Algorithmen: Verfahren, Beispiele, Anwendungen
+            Springer, Berlin; Auflage: 9., überarb. und erw. A. (8. Dezember 2004)
+            Section 4.10.2, page 175
+
+        [3] Hanrath, Wilhelm: Mathematik III / Numerik, Vorlesungsskript zur
+            Veranstaltung im WS 2007/2008
+            Fachhochschule Aachen, 2009-09-19
+            Numerik_01.pdf, downloaded 2011-04-19 via
+            http://www.fh-aachen.de/index.php?id=11424&no_cache=1&file=5016&uid=44191
+            Section 5.3, page 129
+    */
+    void CalculatePeriodic();
 };
 
 //-----------------------------------------------------------------------------
@@ -124,21 +139,32 @@ lcl_SplineCalculation::lcl_SplineCalculation(
           m_nKHigh( rSortedPoints.size() - 1 )
 {
     ::rtl::math::setInf( &m_fLastInterpolatedValue, sal_False );
-
-    // remove points that have equal x-values
-    m_aPoints.erase( ::std::unique( m_aPoints.begin(), m_aPoints.end(),
-                             lcl_EqualsFirstDoubleOfPair< double >() ),
-                     m_aPoints.end() );
     Calculate();
 }
 
+//-----------------------------------------------------------------------------
+
+lcl_SplineCalculation::lcl_SplineCalculation(
+    const tPointVecType & rSortedPoints)
+        : m_aPoints( rSortedPoints ),
+          m_fYp1( 0.0 ),  /*dummy*/
+          m_fYpN( 0.0 ),  /*dummy*/
+          m_nKLow( 0 ),
+          m_nKHigh( rSortedPoints.size() - 1 )
+{
+    ::rtl::math::setInf( &m_fLastInterpolatedValue, sal_False );
+    CalculatePeriodic();
+}
+//-----------------------------------------------------------------------------
+
+
 void lcl_SplineCalculation::Calculate()
 {
     if( m_aPoints.size() <= 1 )
         return;
 
     // n is the last valid index to m_aPoints
-    const tPointVecType::size_type n = m_aPoints.size() - 1;
+    const lcl_tSizeType n = m_aPoints.size() - 1;
     ::std::vector< double > u( n );
     m_aSecDerivY.resize( n + 1, 0.0 );
 
@@ -156,9 +182,9 @@ void lcl_SplineCalculation::Calculate()
             ((( m_aPoints[ 1 ].second - m_aPoints[ 0 ].second ) / xDiff ) - m_fYp1 );
     }
 
-    for( tPointVecType::size_type i = 1; i < n; ++i )
+    for( lcl_tSizeType i = 1; i < n; ++i )
     {
-        ::std::pair< double, double >
+        tPointType
             p_i = m_aPoints[ i ],
             p_im1 = m_aPoints[ i - 1 ],
             p_ip1 = m_aPoints[ i + 1 ];
@@ -196,19 +222,164 @@ void lcl_SplineCalculation::Calculate()
     // note: the algorithm in [1] iterates from n-1 to 0, but as size_type
     // may be (usuall is) an unsigned type, we can not write k >= 0, as this
     // is always true.
-    for( tPointVecType::size_type k = n; k > 0; --k )
+    for( lcl_tSizeType k = n; k > 0; --k )
     {
         ( m_aSecDerivY[ k - 1 ] *= m_aSecDerivY[ k ] ) += u[ k - 1 ];
     }
 }
 
+void lcl_SplineCalculation::CalculatePeriodic()
+{
+    if( m_aPoints.size() <= 1 )
+        return;
+
+    // n is the last valid index to m_aPoints
+    const lcl_tSizeType n = m_aPoints.size() - 1;
+
+    // u is used for vector f in A*c=f in [3], vector a in  Ax=a in [2],
+    // vector z in Rtranspose z = a and Dr=z in [2]
+    ::std::vector< double > u( n + 1, 0.0 );
+
+    // used for vector c in A*c=f and vector x in Ax=a in [2]
+    m_aSecDerivY.resize( n + 1, 0.0 );
+
+    // diagonal of matrix A, used index 1 to n
+    ::std::vector< double > Adiag( n + 1, 0.0 );
+
+    // secondary diagonal of matrix A with index 1 to n-1 and upper right element in A[n]
+    ::std::vector< double > Aupper( n + 1, 0.0 );
+
+    // diagonal of matrix D in A=(R transpose)*D*R in [2], used index 1 to n
+    ::std::vector< double > Ddiag( n+1, 0.0 );
+
+    // right column of matrix R, used index 1 to n-2
+    ::std::vector< double > Rright( n-1, 0.0 );
+
+    // secondary diagonal of matrix R, used index 1 to n-1
+    ::std::vector< double > Rupper( n, 0.0 );
+
+    if (n<4)
+    {
+        if (n==3)
+        {   // special handling of three polynomials, that are four points
+            double xDiff0 = m_aPoints[ 1 ].first - m_aPoints[ 0 ].first ;
+            double xDiff1 = m_aPoints[ 2 ].first - m_aPoints[ 1 ].first ;
+            double xDiff2 = m_aPoints[ 3 ].first - m_aPoints[ 2 ].first ;
+            double xDiff2p1 = xDiff2 + xDiff1;
+            double xDiff0p2 = xDiff0 + xDiff2;
+            double xDiff1p0 = xDiff1 + xDiff0;
+            double fFaktor = 1.5 / (xDiff0*xDiff1 + xDiff1*xDiff2 + xDiff2*xDiff0);
+            double yDiff0 = (m_aPoints[ 1 ].second - m_aPoints[ 0 ].second) / xDiff0;
+            double yDiff1 = (m_aPoints[ 2 ].second - m_aPoints[ 1 ].second) / xDiff1;
+            double yDiff2 = (m_aPoints[ 0 ].second - m_aPoints[ 2 ].second) / xDiff2;
+            m_aSecDerivY[ 1 ] = fFaktor * (yDiff1*xDiff2p1 - yDiff0*xDiff0p2);
+            m_aSecDerivY[ 2 ] = fFaktor * (yDiff2*xDiff0p2 - yDiff1*xDiff1p0);
+            m_aSecDerivY[ 3 ] = fFaktor * (yDiff0*xDiff1p0 - yDiff2*xDiff2p1);
+            m_aSecDerivY[ 0 ] = m_aSecDerivY[ 3 ];
+        }
+        else if (n==2)
+        {
+        // special handling of two polynomials, that are three points
+            double xDiff0 = m_aPoints[ 1 ].first - m_aPoints[ 0 ].first;
+            double xDiff1 = m_aPoints[ 2 ].first - m_aPoints[ 1 ].first;
+            double fHelp = 3.0 * (m_aPoints[ 0 ].second - m_aPoints[ 1 ].second) / (xDiff0*xDiff1);
+            m_aSecDerivY[ 1 ] = fHelp ;
+            m_aSecDerivY[ 2 ] = -fHelp ;
+            m_aSecDerivY[ 0 ] = m_aSecDerivY[ 2 ] ;
+        }
+        else
+        {
+            // should be handled with natural spline, periodic not possible.
+        }
+    }
+    else
+    {
+        double xDiff_i =1.0; // values are dummy;
+        double xDiff_im1 =1.0;
+        double yDiff_i = 1.0;
+        double yDiff_im1 = 1.0;
+        // fill matrix A and fill right side vector u
+        for( lcl_tSizeType i=1; i<n; ++i )
+        {
+            xDiff_im1 = m_aPoints[ i ].first - m_aPoints[ i-1 ].first;
+            xDiff_i = m_aPoints[ i+1 ].first - m_aPoints[ i ].first;
+            yDiff_im1 = (m_aPoints[ i ].second - m_aPoints[ i-1 ].second) / xDiff_im1;
+            yDiff_i = (m_aPoints[ i+1 ].second - m_aPoints[ i ].second) / xDiff_i;
+            Adiag[ i ] = 2 * (xDiff_im1 + xDiff_i);
+            Aupper[ i ] = xDiff_i;
+            u [ i ] = 3 * (yDiff_i - yDiff_im1);
+        }
+        xDiff_im1 = m_aPoints[ n ].first - m_aPoints[ n-1 ].first;
+        xDiff_i = m_aPoints[ 1 ].first - m_aPoints[ 0 ].first;
+        yDiff_im1 = (m_aPoints[ n ].second - m_aPoints[ n-1 ].second) / xDiff_im1;
+        yDiff_i = (m_aPoints[ 1 ].second - m_aPoints[ 0 ].second) / xDiff_i;
+        Adiag[ n ] = 2 * (xDiff_im1 + xDiff_i);
+        Aupper[ n ] = xDiff_i;
+        u [ n ] = 3 * (yDiff_i - yDiff_im1);
+
+        // decomposite A=(R transpose)*D*R
+        Ddiag[1] = Adiag[1];
+        Rupper[1] = Aupper[1] / Ddiag[1];
+        Rright[1] = Aupper[n] / Ddiag[1];
+        for( lcl_tSizeType i=2; i<=n-2; ++i )
+        {
+            Ddiag[i] = Adiag[i] - Aupper[ i-1 ] * Rupper[ i-1 ];
+            Rupper[ i ] = Aupper[ i ] / Ddiag[ i ];
+            Rright[ i ] = - Rright[ i-1 ] * Aupper[ i-1 ] / Ddiag[ i ];
+        }
+        Ddiag[ n-1 ] = Adiag[ n-1 ] - Aupper[ n-2 ] * Rupper[ n-2 ];
+        Rupper[ n-1 ] = ( Aupper[ n-1 ] - Aupper[ n-2 ] * Rright[ n-2] ) / Ddiag[ n-1 ];
+        double fSum = 0.0;
+        for ( lcl_tSizeType i=1; i<=n-2; ++i )
+        {
+            fSum += Ddiag[ i ] * Rright[ i ] * Rright[ i ];
+        }
+        Ddiag[ n ] = Adiag[ n ] - fSum - Ddiag[ n-1 ] * Rupper[ n-1 ] * Rupper[ n-1 ]; // bug in [2]!
+
+        // solve forward (R transpose)*z=u, overwrite u with z
+        for ( lcl_tSizeType i=2; i<=n-1; ++i )
+        {
+            u[ i ] -= u[ i-1 ]* Rupper[ i-1 ];
+        }
+        fSum = 0.0;
+        for ( lcl_tSizeType i=1; i<=n-2; ++i )
+        {
+            fSum += Rright[ i ] * u[ i ];
+        }
+        u[ n ] = u[ n ] - fSum - Rupper[ n - 1] * u[ n-1 ];
+
+        // solve forward D*r=z, z is in u, overwrite u with r
+        for ( lcl_tSizeType i=1; i<=n; ++i )
+        {
+            u[ i ] = u[i] / Ddiag[ i ];
+        }
+
+        // solve backward R*x= r, r is in u
+        m_aSecDerivY[ n ] = u[ n ];
+        m_aSecDerivY[ n-1 ] = u[ n-1 ] - Rupper[ n-1 ] * m_aSecDerivY[ n ];
+        for ( lcl_tSizeType i=n-2; i>=1; --i)
+        {
+            m_aSecDerivY[ i ] = u[ i ] - Rupper[ i ] * m_aSecDerivY[ i+1 ] - Rright[ i ] * m_aSecDerivY[ n ];
+        }
+        // periodic
+        m_aSecDerivY[ 0 ] = m_aSecDerivY[ n ];
+    }
+
+    // adapt m_aSecDerivY for usage in GetInterpolatedValue()
+    for( lcl_tSizeType i = 0; i <= n ; ++i )
+    {
+        m_aSecDerivY[ i ] *= 2.0;
+    }
+
+}
+
 double lcl_SplineCalculation::GetInterpolatedValue( double x )
 {
     OSL_PRECOND( ( m_aPoints[ 0 ].first <= x ) &&
                 ( x <= m_aPoints[ m_aPoints.size() - 1 ].first ),
                 "Trying to extrapolate" );
 
-    const tPointVecType::size_type n = m_aPoints.size() - 1;
+    const lcl_tSizeType n = m_aPoints.size() - 1;
     if( x < m_fLastInterpolatedValue )
     {
         m_nKLow = 0;
@@ -218,7 +389,7 @@ double lcl_SplineCalculation::GetInterpolatedValue( double x )
         // first initialization is done in CTOR
         while( m_nKHigh - m_nKLow > 1 )
         {
-            tPointVecType::size_type k = ( m_nKHigh + m_nKLow ) / 2;
+            lcl_tSizeType k = ( m_nKHigh + m_nKLow ) / 2;
             if( m_aPoints[ k ].first > x )
                 m_nKHigh = k;
             else
@@ -252,63 +423,142 @@ double lcl_SplineCalculation::GetInterpolatedValue( double x )
 
 //-----------------------------------------------------------------------------
 
-//create knot vector for B-spline
-double* createTVector( sal_Int32 n, sal_Int32 k )
-{
-    double* t = new double [n + k + 1];
-    for (sal_Int32 i=0; i<=n+k; i++ )
-    {
-        if(i < k)
-            t[i] = 0;
-        else if(i <= n)
-            t[i] = i-k+1;
+// helper methods for B-spline
+
+// Create parameter t_0 to t_n using the centripetal method with a power of 0.5
+bool createParameterT(const tPointVecType aUniquePoints, double* t)
+{   // precondition: no adjacent identical points
+    // postcondition: 0 = t_0 < t_1 < ... < t_n = 1
+    bool bIsSuccessful = true;
+    const lcl_tSizeType n = aUniquePoints.size() - 1;
+    t[0]=0.0;
+    double dx = 0.0;
+    double dy = 0.0;
+    double fDiffMax = 1.0; //dummy values
+    double fDenominator = 0.0; // initialized for summing up
+    for (lcl_tSizeType i=1; i<=n ; ++i)
+    {   // 4th root(dx^2+dy^2)
+        dx = aUniquePoints[i].first - aUniquePoints[i-1].first;
+        dy = aUniquePoints[i].second - aUniquePoints[i-1].second;
+        // scaling to avoid underflow or overflow
+        fDiffMax = (fabs(dx)>fabs(dy)) ? fabs(dx) : fabs(dy);
+        if (fDiffMax == 0.0)
+        {
+            bIsSuccessful = false;
+            break;
+        }
         else
-            t[i] = n-k+2;
+        {
+            dx /= fDiffMax;
+            dy /= fDiffMax;
+            fDenominator += sqrt(sqrt(dx * dx + dy * dy)) * sqrt(fDiffMax);
+        }
     }
-    return t;
-}
+    if (fDenominator == 0.0)
+    {
+        bIsSuccessful = false;
+    }
+    if (bIsSuccessful)
+    {
+        for (lcl_tSizeType j=1; j<=n ; ++j)
+        {
+            double fNumerator = 0.0;
+            for (lcl_tSizeType i=1; i<=j ; ++i)
+            {
+                dx = aUniquePoints[i].first - aUniquePoints[i-1].first;
+                dy = aUniquePoints[i].second - aUniquePoints[i-1].second;
+                fDiffMax = (abs(dx)>abs(dy)) ? abs(dx) : abs(dy);
+                // same as above, so should not be zero
+                dx /= fDiffMax;
+                dy /= fDiffMax;
+                fNumerator += sqrt(sqrt(dx * dx + dy * dy)) * sqrt(fDiffMax);
+            }
+            t[j] = fNumerator / fDenominator;
 
-//calculate left knot vector
-double TLeft (double x, sal_Int32 i, sal_Int32 k, const double *t )
-{
-    double deltaT = t[i + k - 1] - t[i];
-    return (deltaT == 0.0)
-               ? 0.0
-               : (x - t[i]) / deltaT;
+        }
+        // postcondition check
+        t[n] = 1.0;
+        double fPrevious = 0.0;
+        for (lcl_tSizeType i=1; i <= n && bIsSuccessful ; ++i)
+        {
+            if (fPrevious >= t[i])
+            {
+                bIsSuccessful = false;
+            }
+            else
+            {
+                fPrevious = t[i];
+            }
+        }
+    }
+    return bIsSuccessful;
 }
 
-//calculate right knot vector
-double TRight(double x, sal_Int32 i, sal_Int32 k, const double *t )
-{
-    double deltaT = t[i + k] - t[i + 1];
-    return (deltaT == 0.0)
-               ? 0.0
-               : (t[i + k] - x) / deltaT;
+void createKnotVector(const lcl_tSizeType n, const sal_uInt32 p, double* t, double* u)
+{  // precondition: 0 = t_0 < t_1 < ... < t_n = 1
+        for (lcl_tSizeType j = 0; j <= p; ++j)
+        {
+            u[j] = 0.0;
+        }
+        double fSum = 0.0;
+        for (lcl_tSizeType j = 1; j <= n-p; ++j )
+        {
+            fSum = 0.0;
+            for (lcl_tSizeType i = j; i <= j+p-1; ++i)
+            {
+                fSum += t[i];
+            }
+            u[j+p] = fSum / p ;
+        }
+        for (lcl_tSizeType j = n+1; j <= n+1+p; ++j)
+        {
+            u[j] = 1.0;
+        }
 }
 
-//calculate weight vector
-void BVector(double x, sal_Int32 n, sal_Int32 k, double *b, const double *t)
+void applyNtoParameterT(const lcl_tSizeType i,const double tk,const sal_uInt32 p,const double* u, double* rowN)
 {
-    sal_Int32 i = 0;
-    for( i=0; i<=n+k; i++ )
-        b[i]=0;
+    // get N_p(t_k) recursively, only N_(i-p) till N_(i) are relevant, all other N_# are zero
+    double fRightFactor = 0.0;
+    double fLeftFactor = 0.0;
+
+    // initialize with indicator function degree 0
+    rowN[p] = 1.0; // all others are zero
 
-    sal_Int32 i0 = (sal_Int32)floor(x) + k - 1;
-    b [i0] = 1;
+    // calculate up to degree p
+    for (sal_uInt32 s = 1; s <= p; ++s)
+    {
+        // first element
+        fRightFactor = ( u[i+1] - tk ) / ( u[i+1]- u[i-s+1] );
+        // i-s "true index" - (i-p)"shift" = p-s
+        rowN[p-s] = fRightFactor * rowN[p-s+1];
 
-    for( sal_Int32 j=2; j<=k; j++ )
-        for( i=0; i<=i0; i++ )
-            b[i] = TLeft(x, i, j, t) * b[i] + TRight(x, i, j, t) * b [i + 1];
+        // middle elements
+        for (sal_uInt32 j = s-1; j>=1 ; --j)
+        {
+            fLeftFactor = ( tk - u[i-j] ) / ( u[i-j+s] - u[i-j] ) ;
+            fRightFactor = ( u[i-j+s+1] - tk ) / ( u[i-j+s+1] - u[i-j+1] );
+            // i-j "true index" - (i-p)"shift" = p-j
+            rowN[p-j] = fLeftFactor * rowN[p-j] + fRightFactor *  rowN[p-j+1];
+        }
+
+        // last element
+        fLeftFactor = ( tk - u[i] ) / ( u[i+s] - u[i] );
+        // i "true index" - (i-p)"shift" = p
+        rowN[p] = fLeftFactor * rowN[p];
+    }
 }
 
 } //  anonymous namespace
 
 //-----------------------------------------------------------------------------
 
+// Calculates uniform parametric splines with subinterval length 1,
+// according ODF1.2 part 1, chapter 'chart interpolation'.
 void SplineCalculater::CalculateCubicSplines(
     const drawing::PolyPolygonShape3D& rInput
     , drawing::PolyPolygonShape3D& rResult
-    , sal_Int32 nGranularity )
+    , sal_uInt32 nGranularity )
 {
     OSL_PRECOND( nGranularity > 0, "Granularity is invalid" );
 
@@ -316,7 +566,7 @@ void SplineCalculater::CalculateCubicSplines(
     rResult.SequenceY.realloc(0);
     rResult.SequenceZ.realloc(0);
 
-    sal_Int32 nOuterCount = rInput.SequenceX.getLength();
+    sal_uInt32 nOuterCount = rInput.SequenceX.getLength();
     if( !nOuterCount )
         return;
 
@@ -324,30 +574,23 @@ void SplineCalculater::CalculateCubicSplines(
     rResult.SequenceY.realloc(nOuterCount);
     rResult.SequenceZ.realloc(nOuterCount);
 
-    for( sal_Int32 nOuter = 0; nOuter < nOuterCount; ++nOuter )
+    for( sal_uInt32 nOuter = 0; nOuter < nOuterCount; ++nOuter )
     {
         if( rInput.SequenceX[nOuter].getLength() <= 1 )
             continue; //we need at least two points
 
-        sal_Int32 nMaxIndexPoints = rInput.SequenceX[nOuter].getLength()-1; // is >=1
+        sal_uInt32 nMaxIndexPoints = rInput.SequenceX[nOuter].getLength()-1; // is >=1
         const double* pOldX = rInput.SequenceX[nOuter].getConstArray();
         const double* pOldY = rInput.SequenceY[nOuter].getConstArray();
         const double* pOldZ = rInput.SequenceZ[nOuter].getConstArray();
 
-        // #i13699# The curve gets a parameter and then for each coordinate a
-        // separate spline will be calculated using the parameter as first argument
-        // and the point coordinate as second argument. Therefore the points need
-        // not to be sorted in its x-coordinates. The parameter is sorted by
-        // construction.
-
         ::std::vector < double > aParameter(nMaxIndexPoints+1);
         aParameter[0]=0.0;
-        for( sal_Int32 nIndex=1; nIndex<=nMaxIndexPoints; nIndex++ )
+        for( sal_uInt32 nIndex=1; nIndex<=nMaxIndexPoints; nIndex++ )
         {
-            // The euclidian distance leads to curve loops for functions having single extreme points
-            // use increment of 1 instead
             aParameter[nIndex]=aParameter[nIndex-1]+1;
         }
+
         // Split the calculation to X, Y and Z coordinate
         tPointVecType aInputX;
         aInputX.resize(nMaxIndexPoints+1);
@@ -355,7 +598,7 @@ void SplineCalculater::CalculateCubicSplines(
         aInputY.resize(nMaxIndexPoints+1);
         tPointVecType aInputZ;
         aInputZ.resize(nMaxIndexPoints+1);
-        for (sal_Int32 nN=0;nN<=nMaxIndexPoints; nN++ )
+        for (sal_uInt32 nN=0;nN<=nMaxIndexPoints; nN++ )
         {
           aInputX[ nN ].first=aParameter[nN];
           aInputX[ nN ].second=pOldX[ nN ];
@@ -370,16 +613,20 @@ void SplineCalculater::CalculateCubicSplines(
         double fXDerivation;
         double fYDerivation;
         double fZDerivation;
+        lcl_SplineCalculation* aSplineX;
+        lcl_SplineCalculation* aSplineY;
+        // lcl_SplineCalculation* aSplineZ; the z-coordinates of all points in
+        // a data series are equal. No spline calculation needed, but copy
+        // coordinate to output
+
         if( pOldX[ 0 ] == pOldX[nMaxIndexPoints] &&
             pOldY[ 0 ] == pOldY[nMaxIndexPoints] &&
-            pOldZ[ 0 ] == pOldZ[nMaxIndexPoints] )
-        {
-            // #i101050# avoid a corner in closed lines, which are smoothed by spline
-            // This derivation are special for parameter of kind 0,1,2,3... If you
-            // change generating parameters (see above), then adapt derivations too.)
-            fXDerivation = 0.5 * (pOldX[1]-pOldX[nMaxIndexPoints-1]);
-            fYDerivation = 0.5 * (pOldY[1]-pOldY[nMaxIndexPoints-1]);
-            fZDerivation = 0.5 * (pOldZ[1]-pOldZ[nMaxIndexPoints-1]);
+            pOldZ[ 0 ] == pOldZ[nMaxIndexPoints] &&
+            nMaxIndexPoints >=2 )
+        {   // periodic spline
+            aSplineX = new lcl_SplineCalculation( aInputX) ;
+            aSplineY = new lcl_SplineCalculation( aInputY) ;
+            // aSplineZ = new lcl_SplineCalculation( aInputZ) ;
         }
         else // generate the kind "natural spline"
         {
@@ -388,10 +635,10 @@ void SplineCalculater::CalculateCubicSplines(
             fXDerivation = fInfty;
             fYDerivation = fInfty;
             fZDerivation = fInfty;
+            aSplineX = new lcl_SplineCalculation( aInputX, fXDerivation, fXDerivation );
+            aSplineY = new lcl_SplineCalculation( aInputY, fYDerivation, fYDerivation );
+            // aSplineZ = new lcl_SplineCalculation( aInputZ, fZDerivation, fZDerivation );
         }
-        lcl_SplineCalculation aSplineX( aInputX, fXDerivation, fXDerivation );
-        lcl_SplineCalculation aSplineY( aInputY, fYDerivation, fYDerivation );
-        lcl_SplineCalculation aSplineZ( aInputZ, fZDerivation, fZDerivation );
 
         // fill result polygon with calculated values
         rResult.SequenceX[nOuter].realloc( nMaxIndexPoints*nGranularity + 1);
@@ -402,13 +649,13 @@ void SplineCalculater::CalculateCubicSplines(
         double* pNewY = rResult.SequenceY[nOuter].getArray();
         double* pNewZ = rResult.SequenceZ[nOuter].getArray();
 
-        sal_Int32 nNewPointIndex = 0; // Index in result points
+        sal_uInt32 nNewPointIndex = 0; // Index in result points
         // needed for inner loop
         double    fInc;   // step for intermediate points
-        sal_Int32 nj;     // for loop
+        sal_uInt32 nj;     // for loop
         double    fParam; // a intermediate parameter value
 
-        for( sal_Int32 ni = 0; ni < nMaxIndexPoints; ni++ )
+        for( sal_uInt32 ni = 0; ni < nMaxIndexPoints; ni++ )
         {
             // given point is surely a curve point
             pNewX[nNewPointIndex] = pOldX[ni];
@@ -422,9 +669,10 @@ void SplineCalculater::CalculateCubicSplines(
             {
                 fParam = aParameter[ni] + ( fInc * static_cast< double >( nj ) );
 
-                pNewX[nNewPointIndex]=aSplineX.GetInterpolatedValue( fParam );
-                pNewY[nNewPointIndex]=aSplineY.GetInterpolatedValue( fParam );
-                pNewZ[nNewPointIndex]=aSplineZ.GetInterpolatedValue( fParam );
+                pNewX[nNewPointIndex]=aSplineX->GetInterpolatedValue( fParam );
+                pNewY[nNewPointIndex]=aSplineY->GetInterpolatedValue( fParam );
+                // pNewZ[nNewPointIndex]=aSplineZ->GetInterpolatedValue( fParam );
+                pNewZ[nNewPointIndex] = pOldZ[ni];
                 nNewPointIndex++;
             }
         }
@@ -432,18 +680,34 @@ void SplineCalculater::CalculateCubicSplines(
         pNewX[nNewPointIndex] = pOldX[nMaxIndexPoints];
         pNewY[nNewPointIndex] = pOldY[nMaxIndexPoints];
         pNewZ[nNewPointIndex] = pOldZ[nMaxIndexPoints];
+        delete aSplineX;
+        delete aSplineY;
+        // delete aSplineZ;
     }
 }
 
+
+// The implementation follows closely ODF1.2 spec, chapter chart:interpolation
+// using the same names as in spec as far as possible, without prefix.
+// More details can be found on
+// Dr. C.-K. Shene: CS3621 Introduction to Computing with Geometry Notes
+// Unit 9: Interpolation and Approximation/Curve Global Interpolation
+// Department of Computer Science, Michigan Technological University
+// http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/
+// [last called 2011-05-20]
 void SplineCalculater::CalculateBSplines(
             const ::com::sun::star::drawing::PolyPolygonShape3D& rInput
             , ::com::sun::star::drawing::PolyPolygonShape3D& rResult
-            , sal_Int32 nGranularity
-            , sal_Int32 nDegree )
+            , sal_uInt32 nResolution
+            , sal_uInt32 nDegree )
 {
-    // #issue 72216#
-    // k is the order of the BSpline, nDegree is the degree of its polynoms
-    sal_Int32 k = nDegree + 1;
+    // nResolution is ODF1.2 file format attribut chart:spline-resolution and
+    // ODF1.2 spec variable k. Causion, k is used as index in the spec in addition.
+    // nDegree is ODF1.2 file format attribut chart:spline-order and
+    // ODF1.2 spec variable p
+    OSL_ASSERT( nResolution > 1 );
+    OSL_ASSERT( nDegree >= 1 );
+    sal_uInt32 p = nDegree;
 
     rResult.SequenceX.realloc(0);
     rResult.SequenceY.realloc(0);
@@ -460,74 +724,257 @@ void SplineCalculater::CalculateBSplines(
     for( sal_Int32 nOuter = 0; nOuter < nOuterCount; ++nOuter )
     {
         if( rInput.SequenceX[nOuter].getLength() <= 1 )
-            continue; // need at least 2 control points
-
-        sal_Int32 n = rInput.SequenceX[nOuter].getLength()-1; // maximum index of control points
-
-        double fCurveparam =0.0; // parameter for the curve
-        // 0<= fCurveparam < fMaxCurveparam
-        double fMaxCurveparam = 2.0+ n - k;
-        if (fMaxCurveparam <= 0.0)
-            return; // not enough control points for desired spline order
-
-        if (nGranularity < 1)
-            return; //need at least 1 line for each part beween the control points
+            continue; // need at least 2 points, next piece of the series
 
+        // Copy input to vector of points and remove adjacent double points. The
+        // Z-coordinate is equal for all points in a series and holds the depth
+        // in 3D mode, simple copying is enough.
+        lcl_tSizeType nMaxIndexPoints = rInput.SequenceX[nOuter].getLength()-1; // is >=1
         const double* pOldX = rInput.SequenceX[nOuter].getConstArray();
         const double* pOldY = rInput.SequenceY[nOuter].getConstArray();
         const double* pOldZ = rInput.SequenceZ[nOuter].getConstArray();
+        double fZCoordinate = pOldZ[0];
+        tPointVecType aPointsIn;
+        aPointsIn.resize(nMaxIndexPoints+1);
+        for (lcl_tSizeType i = 0; i <= nMaxIndexPoints; ++i )
+        {
+          aPointsIn[ i ].first = pOldX[i];
+          aPointsIn[ i ].second = pOldY[i];
+        }
+        aPointsIn.erase( ::std::unique( aPointsIn.begin(), aPointsIn.end()),
+                     aPointsIn.end() );
+
+        // n is the last valid index to the reduced aPointsIn
+        // There are n+1 valid data points.
+        const lcl_tSizeType n = aPointsIn.size() - 1;
+        if (n < 1 || p > n)
+            continue; // need at least 2 points, degree p needs at least n+1 points
+                      // next piece of series
+
+        double* t = new double [n+1];
+        if (!createParameterT(aPointsIn, t))
+        {
+            delete[] t;
+            continue; // next piece of series
+        }
 
-        // keep this amount of steps to go well with old version
-        sal_Int32 nNewSectorCount = nGranularity * n;
-        double fCurveStep = fMaxCurveparam/static_cast< double >(nNewSectorCount);
-
-        double *b       = new double [n + k + 1]; // values of blending functions
-
-        const double* t = createTVector(n, k); // knot vector
-
-        rResult.SequenceX[nOuter].realloc(nNewSectorCount+1);
-        rResult.SequenceY[nOuter].realloc(nNewSectorCount+1);
-        rResult.SequenceZ[nOuter].realloc(nNewSectorCount+1);
-        double* pNewX = rResult.SequenceX[nOuter].getArray();
-        double* pNewY = rResult.SequenceY[nOuter].getArray();
-        double* pNewZ = rResult.SequenceZ[nOuter].getArray();
-
-        // variables needed inside loop, when calculating one point of output
-        sal_Int32 nPointIndex =0; //index of given contol points
-        double fX=0.0;
-        double fY=0.0;
-        double fZ=0.0; //coordinates of a new BSpline point
-
-        for(sal_Int32 nNewSector=0; nNewSector<nNewSectorCount; nNewSector++)
-        { // in first looping fCurveparam has value 0.0
-
-            // Calculate the values of the blending functions for actual curve parameter
-            BVector(fCurveparam, n, k, b, t);
-
-            // output point(fCurveparam) = sum over {input point * value of blending function}
-            fX = 0.0;
-            fY = 0.0;
-            fZ = 0.0;
-            for (nPointIndex=0;nPointIndex<=n;nPointIndex++)
+        lcl_tSizeType m = n + p + 1;
+        double* u = new double [m+1];
+        createKnotVector(n, p, t, u);
+
+        // The matrix N contains the B-spline basis functions applied to parameters.
+        // In each row only p+1 adjacent elements are non-zero. The starting
+        // column in a higher row is equal or greater than in the lower row.
+        // To store this matrix the non-zero elements are shifted to column 0
+        // and the amount of shifting is remembered in an array.
+        double** aMatN = new double*[n+1];
+        for (lcl_tSizeType row = 0; row <=n; ++row)
+        {
+            aMatN[row] = new double[p+1];
+            for (sal_uInt32 col = 0; col <= p; ++col)
+            aMatN[row][col] = 0.0;
+        }
+        lcl_tSizeType* aShift = new lcl_tSizeType[n+1];
+        aMatN[0][0] = 1.0; //all others are zero
+        aShift[0] = 0;
+        aMatN[n][0] = 1.0;
+        aShift[n] = n;
+        for (lcl_tSizeType k = 1; k<=n-1; ++k)
+        { // all basis functions are applied to t_k,
+            // results are elements in row k in matrix N
+
+            // find the one interval with u_i <= t_k < u_(i+1)
+            // remember u_0 = ... = u_p = 0.0 and u_(m-p) = ... u_m = 1.0 and 0<t_k<1
+            lcl_tSizeType i = p;
+            while (!(u[i] <= t[k] && t[k] < u[i+1]))
             {
-                fX +=pOldX[nPointIndex]*b[nPointIndex];
-                fY +=pOldY[nPointIndex]*b[nPointIndex];
-                fZ +=pOldZ[nPointIndex]*b[nPointIndex];
+                ++i;
             }
-            pNewX[nNewSector] = fX;
-            pNewY[nNewSector] = fY;
-            pNewZ[nNewSector] = fZ;
 
-            fCurveparam += fCurveStep; //for next looping
+            // index in reduced matrix aMatN = (index in full matrix N) - (i-p)
+            aShift[k] = i - p;
+
+            applyNtoParameterT(i, t[k], p, u, aMatN[k]);
+        } // next row k
+
+        // Get matrix C of control points from the matrix equation aMatN * C = aPointsIn
+        // aPointsIn is overwritten with C.
+        // Gaussian elimination is possible without pivoting, see reference
+        lcl_tSizeType r = 0; // true row index
+        lcl_tSizeType c = 0; // true column index
+        double fDivisor = 1.0; // used for diagonal element
+        double fEliminate = 1.0; // used for the element, that will become zero
+        double fHelp;
+        tPointType aHelp;
+        lcl_tSizeType nHelp; // used in triangle change
+        bool bIsSuccessful = true;
+        for (c = 0 ; c <= n && bIsSuccessful; ++c)
+        {
+            // search for first non-zero downwards
+            r = c;
+            while ( aMatN[r][c-aShift[r]] == 0 &&  r < n)
+            {
+                ++r;
+            }
+            if (aMatN[r][c-aShift[r]] == 0.0)
+            {
+                // Matrix N is singular, although this is mathematically impossible
+                bIsSuccessful = false;
+            }
+            else
+            {
+                // exchange total row r with total row c if necessary
+                if (r != c)
+                {
+                    for ( sal_uInt32 i = 0; i <= p ; ++i)
+                    {
+                        fHelp = aMatN[r][i];
+                        aMatN[r][i] = aMatN[c][i];
+                        aMatN[c][i] = fHelp;
+                    }
+                    aHelp = aPointsIn[r];
+                    aPointsIn[r] = aPointsIn[c];
+                    aPointsIn[c] = aHelp;
+                    nHelp = aShift[r];
+                    aShift[r] = aShift[c];
+                    aShift[c] = nHelp;
+                }
+
+                // divide row c, so that element(c,c) becomes 1
+                fDivisor = aMatN[c][c-aShift[c]]; // not zero, see above
+                for (sal_uInt32 i = 0; i <= p; ++i)
+                {
+                    aMatN[c][i] /= fDivisor;
+                }
+                aPointsIn[c].first /= fDivisor;
+                aPointsIn[c].second /= fDivisor;
+
+                // eliminate forward, examine row c+1 to n-1 (worst case)
+                // stop if first non-zero element in row has an higher column as c
+                // look at nShift for that, elements in nShift are equal or increasing
+                for ( r = c+1; aShift[r]<=c && r < n; ++r)
+                {
+                    fEliminate = aMatN[r][0];
+                    if (fEliminate != 0.0) // else accidentally zero, nothing to do
+                    {
+                        for (sal_uInt32 i = 1; i <= p; ++i)
+                        {
+                            aMatN[r][i-1] = aMatN[r][i] - fEliminate * aMatN[c][i];
+                        }
+                        aMatN[r][p]=0;
+                        aPointsIn[r].first -= fEliminate * aPointsIn[c].first;
+                        aPointsIn[r].second -= fEliminate * aPointsIn[c].second;
+                        ++aShift[r];
+                    }
+                }
+            }
+        }// upper triangle form is reached
+        if( bIsSuccessful)
+        {
+            // eliminate backwards, begin with last column
+            for (lcl_tSizeType cc = n; cc >= 1; --cc )
+            {
+                // In row cc the diagonal element(cc,cc) == 1 and all elements left from
+                // diagonal are zero and do not influence other rows.
+                // Full matrix N has semibandwidth < p, therefore element(r,c) is
+                // zero, if abs(r-cc)>=p.  abs(r-cc)=cc-r, because r<cc.
+                r = cc - 1;
+                while ( r !=0 && cc-r < p )
+                {
+                    fEliminate = aMatN[r][ cc - aShift[r] ];
+                    if ( fEliminate != 0.0) // else element is accidentically zero, no action needed
+                    {
+                        // row r -= fEliminate * row cc only relevant for right side
+                        aMatN[r][cc - aShift[r]] = 0.0;
+                        aPointsIn[r].first -= fEliminate * aPointsIn[cc].first;
+                        aPointsIn[r].second -= fEliminate * aPointsIn[cc].second;
+                    }
+                    --r;
+                }
+            }
+        }   // aPointsIn contains the control points now.
+        if (bIsSuccessful)
+        {
+            // calculate the intermediate points according given resolution
+            // using deBoor-Cox algorithm
+            lcl_tSizeType nNewSize = nResolution * n + 1;
+            rResult.SequenceX[nOuter].realloc(nNewSize);
+            rResult.SequenceY[nOuter].realloc(nNewSize);
+            rResult.SequenceZ[nOuter].realloc(nNewSize);
+            double* pNewX = rResult.SequenceX[nOuter].getArray();
+            double* pNewY = rResult.SequenceY[nOuter].getArray();
+            double* pNewZ = rResult.SequenceZ[nOuter].getArray();
+            pNewX[0] = aPointsIn[0].first;
+            pNewY[0] = aPointsIn[0].second;
+            pNewZ[0] = fZCoordinate; // Precondition: z-coordinates of all points of a series are equal
+            pNewX[nNewSize -1 ] = aPointsIn[n].first;
+            pNewY[nNewSize -1 ] = aPointsIn[n].second;
+            pNewZ[nNewSize -1 ] = fZCoordinate;
+            double* aP = new double[m+1];
+            lcl_tSizeType nLow = 0;
+            for ( lcl_tSizeType nTIndex = 0; nTIndex <= n-1; ++nTIndex)
+            {
+                for (sal_uInt32 nResolutionStep = 1;
+                     nResolutionStep <= nResolution && !( nTIndex == n-1 && nResolutionStep == nResolution);
+                     ++nResolutionStep)
+                {
+                    lcl_tSizeType nNewIndex = nTIndex * nResolution + nResolutionStep;
+                    double ux = t[nTIndex] + nResolutionStep * ( t[nTIndex+1] - t[nTIndex]) /nResolution;
+
+                    // get index nLow, so that u[nLow]<= ux < u[nLow +1]
+                    // continue from previous nLow
+                    while ( u[nLow] <= ux)
+                    {
+                        ++nLow;
+                    }
+                    --nLow;
+
+                    // x-coordinate
+                    for (lcl_tSizeType i = nLow-p; i <= nLow; ++i)
+                    {
+                        aP[i] = aPointsIn[i].first;
+                    }
+                    for (sal_uInt32 lcl_Degree = 1; lcl_Degree <= p; ++lcl_Degree)
+                    {
+                        double fFactor = 0.0;
+                        for (lcl_tSizeType i = nLow; i >= nLow + lcl_Degree - p; --i)
+                        {
+                            fFactor = ( ux - u[i] ) / ( u[i+p+1-lcl_Degree] - u[i]);
+                            aP[i] = (1 - fFactor)* aP[i-1] + fFactor * aP[i];
+                        }
+                    }
+                    pNewX[nNewIndex] = aP[nLow];
+
+                    // y-coordinate
+                    for (lcl_tSizeType i = nLow - p; i <= nLow; ++i)
+                    {
+                        aP[i] = aPointsIn[i].second;
+                    }
+                    for (sal_uInt32 lcl_Degree = 1; lcl_Degree <= p; ++lcl_Degree)
+                    {
+                        double fFactor = 0.0;
+                        for (lcl_tSizeType i = nLow; i >= nLow +lcl_Degree - p; --i)
+                        {
+                            fFactor = ( ux - u[i] ) / ( u[i+p+1-lcl_Degree] - u[i]);
+                            aP[i] = (1 - fFactor)* aP[i-1] + fFactor * aP[i];
+                        }
+                    }
+                    pNewY[nNewIndex] = aP[nLow];
+                    pNewZ[nNewIndex] = fZCoordinate;
+                }
+            }
+            delete[] aP;
         }
-        // add last control point to BSpline curve
-        pNewX[nNewSectorCount] = pOldX[n];
-        pNewY[nNewSectorCount] = pOldY[n];
-        pNewZ[nNewSectorCount] = pOldZ[n];
-
+        delete[] aShift;
+        for (lcl_tSizeType row = 0; row <=n; ++row)
+        {
+            delete[] aMatN[row];
+        }
+        delete[] aMatN;
+        delete[] u;
         delete[] t;
-        delete[] b;
-    }
+
+    } // next piece of the series
 }
 
 //.............................................................................
diff --git a/chart2/source/view/charttypes/Splines.hxx b/chart2/source/view/charttypes/Splines.hxx
index 0e8206fcade1..4f4d10483f0d 100644
--- a/chart2/source/view/charttypes/Splines.hxx
+++ b/chart2/source/view/charttypes/Splines.hxx
@@ -46,13 +46,13 @@ public:
     static void CalculateCubicSplines(
             const ::com::sun::star::drawing::PolyPolygonShape3D& rPoints
             , ::com::sun::star::drawing::PolyPolygonShape3D& rResult
-            , sal_Int32 nGranularity );
+            , sal_uInt32 nGranularity );
 
     static void CalculateBSplines(
             const ::com::sun::star::drawing::PolyPolygonShape3D& rPoints
             , ::com::sun::star::drawing::PolyPolygonShape3D& rResult
-            , sal_Int32 nGranularity
-            , sal_Int32 nSplineDepth );
+            , sal_uInt32 nGranularity
+            , sal_uInt32 nSplineDepth );
 };