kicad/dxflib_qcad/tinyspline_lib/tinyspline.c

/*
 *  The MIT License (MIT)
 *
 *  Copyright (c) 2016 Marcel Steinbeck
 *
 *  Permission is hereby granted, free of charge, to any person obtaining a copy
 *  of this software and associated documentation files (the "Software"), to deal
 *  in the Software without restriction, including without limitation the rights
 *  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 *  copies of the Software, and to permit persons to whom the Software is
 *  furnished to do so, subject to the following conditions:
 *
 *  The above copyright notice and this permission notice shall be included in all
 *  copies or substantial portions of the Software.
 *
 *  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 *  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 *  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 *  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 *  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 *  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 *  SOFTWARE.
 */

#include "tinyspline.h"

#include <stdlib.h>     /* malloc, free */
#include <math.h>       /* fabs, sqrt */
#include <string.h>     /* memcpy, memmove, strcmp */
#include <setjmp.h>     /* setjmp, longjmp */


/********************************************************
*                                                       *
* Error handling                                        *
*                                                       *
********************************************************/
#define TRY( x, y ) y = (tsError) setjmp( x ); if( y == 0 ) {
#define CATCH   } \
    else {
#define ETRY    }


/********************************************************
*                                                       *
* Internal functions                                    *
*                                                       *
********************************************************/
void ts_internal_deboornet_copy( const tsDeBoorNet* original,
        tsDeBoorNet* copy, jmp_buf buf
        )
{
    const size_t dim = original->dim;
    const size_t n_points   = original->n_points;
    const size_t sof_f  = sizeof(tsReal);
    const size_t sof_p  = n_points * dim * sof_f;

    if( original == copy )
        return;

    copy->u = original->u;
    copy->k = original->k;
    copy->s = original->s;
    copy->h = original->h;
    copy->dim = dim;
    copy->n_points = n_points;
    copy->points = (tsReal*) malloc( sof_p );

    if( copy->points == NULL )
        longjmp( buf, TS_MALLOC );

    memcpy( copy->points, original->points, sof_p );
    copy->result = copy->points + (n_points - 1) * dim;
}


void ts_internal_bspline_find_u( const tsBSpline* bspline, const tsReal u,
        size_t* k, size_t* s, jmp_buf buf
        )
{
    const size_t deg = bspline->deg;
    const size_t order = bspline->order;
    const size_t n_knots = bspline->n_knots;

    *k = *s = 0;

    for( ; *k < n_knots; (*k)++ )
    {
        const tsReal uk = bspline->knots[*k];

        if( ts_fequals( u, uk ) )
        {
            (*s)++;
        }
        else if( u < uk )
        {
            break;
        }
    }

    /* keep in mind that currently k is k+1 */
    if( *s > order )
        longjmp( buf, TS_MULTIPLICITY );

    if( *k <= deg )                /* u < u_min */
        longjmp( buf, TS_U_UNDEFINED );

    if( *k == n_knots && *s == 0 )    /* u > u_last */
        longjmp( buf, TS_U_UNDEFINED );

    if( *k > n_knots - deg + *s - 1 )    /* u > u_max */
        longjmp( buf, TS_U_UNDEFINED );

    (*k)--;    /* k+1 - 1 will never underflow */
}


void ts_internal_bspline_copy( const tsBSpline* original,
        tsBSpline* copy, jmp_buf buf
        )
{
    const size_t dim = original->dim;
    const size_t sof_f = sizeof(tsReal);
    const size_t n_ctrlp    = original->n_ctrlp;
    const size_t n_knots    = original->n_knots;
    const size_t sof_ck     = (n_ctrlp * dim + n_knots) * sof_f;

    /* Nothing to do here. */
    if( original == copy )
        return;

    copy->deg = original->deg;
    copy->order = original->order;
    copy->dim = original->dim;
    copy->n_ctrlp = original->n_ctrlp;
    copy->n_knots = original->n_knots;
    copy->ctrlp = (tsReal*) malloc( sof_ck );

    if( copy->ctrlp == NULL )
        longjmp( buf, TS_MALLOC );

    memcpy( copy->ctrlp, original->ctrlp, sof_ck );
    copy->knots = copy->ctrlp + n_ctrlp * dim;
}


void ts_internal_bspline_fill_knots( const tsBSpline* original, const tsBSplineType type,
        const tsReal min, const tsReal max,
        tsBSpline* result, jmp_buf buf
        )
{
    const size_t n_knots = original->n_knots;
    const size_t deg = original->deg;
    const size_t order = deg + 1;   /* Using deg+1 instead of original->order
                                     * ensures order >= 1. */
    tsReal fac;                     /* The factor used to calculate the knot values. */
    size_t i;                       /* Used in for loops. */

    /* order >= 1 implies 2*order >= 2 implies n_knots >= 2 */
    if( n_knots < 2 * order )
        longjmp( buf, TS_DEG_GE_NCTRLP );

    if( type == TS_BEZIERS && n_knots % order != 0 )
        longjmp( buf, TS_NUM_KNOTS );

    if( min > max || ts_fequals( min, max ) )
        longjmp( buf, TS_KNOTS_DECR );

    /* copy spline even if type is TS_NONE */
    ts_internal_bspline_copy( original, result, buf );

    if( type == TS_OPENED )
    {
        /* ensures that the first knot value is exactly \min */
        result->knots[0] = min;             /* n_knots >= 2 */

        fac = (max - min) / (n_knots - 1);  /* n_knots >= 2 */

        for( i = 1; i < n_knots - 1; i++ )
            result->knots[i] = min + i * fac;

        /* ensure that the last knot value is exactly \max */
        result->knots[i] = max;
    }
    else if( type == TS_CLAMPED )
    {
        /* n_knots >= 2*order == 2*(deg+1) == 2*deg + 2 > 2*deg - 1 */
        fac = (max - min) / (n_knots - 2 * deg - 1);

        ts_arr_fill( result->knots, order, min );

        for( i = order; i < n_knots - order; i++ )
            result->knots[i] = min + (i - deg) * fac;

        ts_arr_fill( result->knots + i, order, max );
    }
    else if( type == TS_BEZIERS )
    {
        /* n_knots >= 2*order implies n_knots/order >= 2 */
        fac = (max - min) / (n_knots / order - 1);

        ts_arr_fill( result->knots, order, min );

        for( i = order; i < n_knots - order; i += order )
            ts_arr_fill( result->knots + i, order, min + (i / order) * fac );

        ts_arr_fill( result->knots + i, order, max );
    }
}


void ts_internal_bspline_new( const size_t n_ctrlp, const size_t dim, const size_t deg,
        const tsBSplineType type, tsBSpline* bspline, jmp_buf buf
        )
{
    const size_t order = deg + 1;
    const size_t n_knots    = n_ctrlp + order;
    const size_t sof_f  = sizeof(tsReal);
    const size_t sof_ck = (n_ctrlp * dim + n_knots) * sof_f;
    tsError e;
    jmp_buf b;

    if( dim < 1 )
        longjmp( buf, TS_DIM_ZERO );

    if( deg >= n_ctrlp )
        longjmp( buf, TS_DEG_GE_NCTRLP );

    bspline->deg = deg;
    bspline->order = order;
    bspline->dim = dim;
    bspline->n_ctrlp = n_ctrlp;
    bspline->n_knots = n_knots;
    bspline->ctrlp = (tsReal*) malloc( sof_ck );

    if( bspline->ctrlp == NULL )
        longjmp( buf, TS_MALLOC );

    bspline->knots = bspline->ctrlp + n_ctrlp * dim;

    TRY( b, e )
    ts_internal_bspline_fill_knots( bspline, type, 0.f, 1.f, bspline, b );
    CATCH free( bspline->ctrlp );

    longjmp( buf, e );
    ETRY
}


void ts_internal_bspline_resize( const tsBSpline* bspline, const int n, const int back,
        tsBSpline* resized, jmp_buf buf
        )
{
    const size_t deg    = bspline->deg;
    const size_t dim    = bspline->dim;
    const size_t sof_f = sizeof(tsReal);
    const size_t sof_c = dim * sof_f;

    const size_t n_ctrlp = bspline->n_ctrlp;
    const size_t n_knots = bspline->n_knots;
    const size_t nn_ctrlp = n_ctrlp + n;                        /* The new length of ctrlp. */
    const size_t nn_knots = n_knots + n;                        /* The new length of knots. */
    const size_t sof_ncnk = (nn_ctrlp * dim + nn_knots) * sof_f;
    const size_t min_n_ctrlp    = n < 0 ? nn_ctrlp : n_ctrlp;   /* The minimum of
                                                                 * the control points old and new size. */
    const size_t min_n_knots    = n < 0 ? nn_knots : n_knots;   /* the minimum of
                                                                 * the knots old and new size. */

    tsReal* from_ctrlp  = bspline->ctrlp;
    tsReal* from_knots  = bspline->knots;
    tsReal* to_ctrlp    = NULL;
    tsReal* to_knots    = NULL;

    /* If n is 0 the spline must not be resized. */
    if( n == 0 )
    {
        ts_internal_bspline_copy( bspline, resized, buf );
        return;
    }

    if( bspline != resized )
    {
        ts_internal_bspline_new( nn_ctrlp, dim, deg, TS_NONE, resized, buf );
        to_ctrlp = resized->ctrlp;
        to_knots = resized->knots;
    }
    else
    {
        if( nn_ctrlp <= deg )
            longjmp( buf, TS_DEG_GE_NCTRLP );

        to_ctrlp = (tsReal*) malloc( sof_ncnk );

        if( to_ctrlp == NULL )
            longjmp( buf, TS_MALLOC );

        to_knots = to_ctrlp + nn_ctrlp * dim;
    }

    /* Copy control points and knots. */
    if( !back && n < 0 )
    {
        memcpy( to_ctrlp, from_ctrlp - n * dim, min_n_ctrlp * sof_c );
        memcpy( to_knots, from_knots - n, min_n_knots * sof_f );
    }
    else if( !back && n > 0 )
    {
        memcpy( to_ctrlp + n * dim, from_ctrlp, min_n_ctrlp * sof_c );
        memcpy( to_knots + n, from_knots, min_n_knots * sof_f );
    }
    else
    {
        /* n != 0 implies back == true */
        memcpy( to_ctrlp, from_ctrlp, min_n_ctrlp * sof_c );
        memcpy( to_knots, from_knots, min_n_knots * sof_f );
    }

    /* Cleanup if necessary. */
    if( bspline == resized )
    {
        /* free old memory */
        free( from_ctrlp );
        /* assign new values */
        resized->ctrlp  = to_ctrlp;
        resized->knots  = to_knots;
        resized->n_ctrlp = nn_ctrlp;
        resized->n_knots = nn_knots;
    }
}


void ts_internal_bspline_insert_knot( const tsBSpline* bspline,
        const tsDeBoorNet* deBoorNet,
        const size_t n,
        tsBSpline* result,
        jmp_buf buf
        )
{
    const size_t deg    = bspline->deg;
    const size_t dim    = bspline->dim;
    const size_t k = deBoorNet->k;
    const size_t sof_f = sizeof(tsReal);
    const size_t sof_c = dim * sof_f;
    size_t N;       /* The number of affected control points. */
    tsReal* from;   /* The pointer to copy the values from. */
    tsReal* to;     /* The pointer to copy the values to. */
    int stride;     /* The stride of the next pointer to copy. Will be negative
                     * later on, thus use int. */
    size_t i;       /* Used in for loops. */

    if( deBoorNet->s + n > bspline->order )
    {
        longjmp( buf, TS_MULTIPLICITY );
    }

    /* Use ::ts_bspline_resize even if \n is 0 to copy
     * the spline if necessary. */
    ts_internal_bspline_resize( bspline, (int) n, 1, result, buf );

    if( n == 0 )    /* Nothing to insert. */
        return;

    N = deBoorNet->h + 1;    /* n > 0 implies s <= deg implies a regular evaluation
                              * implies h+1 is valid. */

    /* 1. Copy all necessary control points and knots from
     *    the original B-Spline.
     * 2. Copy all necessary control points and knots from
     *    the de Boor net. */

    /* 1.
     *
     * a) Copy left hand side control points from original b-spline.
     * b) Copy right hand side control points from original b-spline.
     * c) Copy left hand side knots from original b-spline.
     * d) Copy right hand side knots form original b-spline. */
    /* copy control points */
    memmove( result->ctrlp, bspline->ctrlp, (k - deg) * sof_c );            /* a) */
    from = bspline->ctrlp + dim * (k - deg + N);
    to = result->ctrlp + dim * (k - deg + N + n);                           /* n >= 0 implies to >= from */
    memmove( to, from, ( result->n_ctrlp - n - (k - deg + N) ) * sof_c );   /* b) */
    /* copy knots */
    memmove( result->knots, bspline->knots, (k + 1) * sof_f );              /* c) */
    from = bspline->knots + k + 1;
    to = result->knots + k + 1 + n;                                         /* n >= 0 implies to >= from */
    memmove( to, from, ( result->n_knots - n - (k + 1) ) * sof_f );         /* d) */

    /* 2.
     *
     * a) Copy left hand side control points from de boor net.
     * b) Copy middle part control points from de boor net.
     * c) Copy right hand side control points from de boor net.
     * d) Insert knots with u_k. */
    from = deBoorNet->points;
    to = result->ctrlp + (k - deg) * dim;
    stride = (int) (N * dim);

    /* copy control points */
    for( i = 0; i < n; i++ )    /* a) */
    {
        memcpy( to, from, sof_c );
        from += stride;
        to += dim;
        stride -= (int) dim;
    }

    memcpy( to, from, (N - n) * sof_c );    /* b) */

    from -= dim;
    to += (N - n) * dim;
    stride = -(int) (N - n + 1) * (int) dim;    /* N = h+1 with h = deg-s
                                                * (ts_internal_bspline_evaluate) implies N = deg-s+1 = order-s.
                                                * n <= order-s implies N-n+1 >= order-s - order-s + 1 = 1. Thus,
                                                * -(int)(N-n+1) <= -1. */

    for( i = 0; i < n; i++ )                    /* c) */
    {
        memcpy( to, from, sof_c );
        from += stride;
        stride -= (int) dim;
        to += dim;
    }

    /* copy knots */
    to = result->knots + k + 1;

    for( i = 0; i < n; i++ )    /* d) */
    {
        *to = deBoorNet->u;
        to++;
    }
}


void ts_internal_bspline_evaluate( const tsBSpline* bspline, const tsReal u,
        tsDeBoorNet* deBoorNet, jmp_buf buf
        )
{
    const size_t deg = bspline->deg;
    const size_t order = bspline->order;
    const size_t dim = bspline->dim;
    const size_t sof_c = dim * sizeof(tsReal); /* The size of a single
                                                * control points.*/
    size_t k;
    size_t s;
    tsReal uk;          /* The actual used u. */
    size_t from;        /* An offset used to copy values. */
    size_t fst;         /* The first affected control point, inclusive. */
    size_t lst;         /* The last affected control point, inclusive. */
    size_t N;           /* The number of affected control points. */
    /* the following indices are used to create the DeBoor net. */
    size_t lidx;        /* The current left index. */
    size_t ridx;        /* The current right index. */
    size_t tidx;        /* The current to index. */
    size_t r, i, d;     /* Used in for loop. */
    tsReal ui;          /* The knot value at index i. */
    tsReal a, a_hat;    /* The weighting factors of the control points. */

    /* Setup the net with its default values. */
    ts_deboornet_default( deBoorNet );

    /* 1. Find index k such that u is in between [u_k, u_k+1).
     * 2. Setup already known values.
     * 3. Decide by multiplicity of u how to calculate point P(u). */

    /* 1. */
    ts_internal_bspline_find_u( bspline, u, &k, &s, buf );
    deBoorNet->k = k;
    deBoorNet->s = s;

    /* 2. */
    uk = bspline->knots[k];                         /* Ensures that with any */
    deBoorNet->u = ts_fequals( u, uk ) ? uk : u;    /* tsReal precision the
                                                     * knot vector stays valid. */
    deBoorNet->h = deg < s ? 0 : deg - s;           /* prevent underflow */
    deBoorNet->dim = dim;

    /* 3. (by 1. s <= order)
     *
     * 3a) Check for s = order.
     *     Take the two points k-s and k-s + 1. If one of
     *     them doesn't exist, take only the other.
     * 3b) Use de boor algorithm to find point P(u). */
    if( s == order )
    {
        /* only one of the two control points exists */
        if( k == deg                        /* only the first control point */
            || k == bspline->n_knots - 1 )  /* only the last control point */
        {
            deBoorNet->points = (tsReal*) malloc( sof_c );

            if( deBoorNet->points == NULL )
                longjmp( buf, TS_MALLOC );

            deBoorNet->result = deBoorNet->points;
            deBoorNet->n_points = 1;
            from = k == deg ? 0 : (k - s) * dim;
            memcpy( deBoorNet->points, bspline->ctrlp + from, sof_c );
        }
        else
        {
            deBoorNet->points = (tsReal*) malloc( 2 * sof_c );

            if( deBoorNet->points == NULL )
                longjmp( buf, TS_MALLOC );

            deBoorNet->result = deBoorNet->points + dim;
            deBoorNet->n_points = 2;
            from = (k - s) * dim;
            memcpy( deBoorNet->points, bspline->ctrlp + from, 2 * sof_c );
        }
    }
    else                                                        /* by 3a) s <= deg (order = deg+1) */
    {
        fst = k - deg;                                          /* by 1. k >= deg */
        lst = k - s;                                            /* s <= deg <= k */
        N = lst - fst + 1;                                      /* lst <= fst implies N >= 1 */

        deBoorNet->n_points = (size_t) (N * (N + 1) * 0.5f);    /* always fits */
        deBoorNet->points = (tsReal*) malloc( deBoorNet->n_points * sof_c );

        if( deBoorNet->points == NULL )
            longjmp( buf, TS_MALLOC );

        deBoorNet->result = deBoorNet->points + (deBoorNet->n_points - 1) * dim;

        /* copy initial values to output */
        memcpy( deBoorNet->points, bspline->ctrlp + fst * dim, N * sof_c );

        lidx = 0;
        ridx = dim;
        tidx = N * dim; /* N >= 1 implies tidx > 0 */
        r = 1;

        for( ; r <= deBoorNet->h; r++ )
        {
            i = fst + r;

            for( ; i <= lst; i++ )
            {
                ui  = bspline->knots[i];
                a   = (deBoorNet->u - ui) / (bspline->knots[i + deg - r + 1] - ui);
                a_hat = 1.f - a;

                for( d = 0; d < dim; d++ )
                {
                    deBoorNet->points[tidx++] =
                        a_hat * deBoorNet->points[lidx++] +
                        a * deBoorNet->points[ridx++];
                }
            }

            lidx += dim;
            ridx += dim;
        }
    }
}


void ts_internal_bspline_split( const tsBSpline* bspline, const tsReal u,
        tsBSpline* split, size_t* k, jmp_buf buf
        )
{
    tsDeBoorNet net;
    tsError e;
    jmp_buf b;

    TRY( b, e )
    ts_internal_bspline_evaluate( bspline, u, &net, b );

    if( net.s == bspline->order )
    {
        ts_internal_bspline_copy( bspline, split, b );
        *k = net.k;
    }
    else
    {
        ts_internal_bspline_insert_knot(
                bspline, &net, net.h + 1, split, b );
        *k = net.k + net.h + 1;
    }

    CATCH
    * k = 0;
    ETRY ts_deboornet_free(& net );

    if( e < 0 )
        longjmp( buf, e );
}


void ts_internal_bspline_thomas_algorithm( const tsReal* points, const size_t n, const size_t dim,
        tsReal* output, jmp_buf buf
        )
{
    const size_t sof_f = sizeof(tsReal);    /* The size of a tsReal. */
    const size_t sof_c = dim * sof_f;       /* The size of a single control point. */
    size_t len_m;                           /* The length m. */
    tsReal* m;                              /* The array of weights. */
    size_t  lst;                            /* The index of the last control point in \points. */
    size_t  i, d;                           /* Used in for loops. */
    size_t  j, k, l;                        /* Used as temporary indices. */

    /* input validation */
    if( dim == 0 )
        longjmp( buf, TS_DIM_ZERO );

    if( n == 0 )
        longjmp( buf, TS_DEG_GE_NCTRLP );

    if( n <= 2 )
    {
        memcpy( output, points, n * sof_c );
        return;
    }

    /* In the following n >= 3 applies... */
    len_m = n - 2;          /* ... len_m >= 1 */
    lst = (n - 1) * dim;    /* ... lst >= 2*dim */

    /* m_0 = 1/4, m_{k+1} = 1/(4-m_k), for k = 0,...,n-2 */
    m = (tsReal*) malloc( len_m * sof_f );

    if( m == NULL )
        longjmp( buf, TS_MALLOC );

    m[0] = 0.25f;

    for( i = 1; i < len_m; i++ )
        m[i] = 1.f / (4 - m[i - 1]);

    /* forward sweep */
    ts_arr_fill( output, n * dim, 0.f );
    memcpy( output, points, sof_c );
    memcpy( output + lst, points + lst, sof_c );

    for( d = 0; d < dim; d++ )
    {
        k = dim + d;
        output[k] = 6 * points[k];
        output[k] -= points[d];
    }

    for( i = 2; i <= n - 2; i++ )
    {
        for( d = 0; d < dim; d++ )
        {
            j = (i - 1) * dim + d;
            k = i * dim + d;
            l = (i + 1) * dim + d;
            output[k] = 6 * points[k];
            output[k] -= output[l];
            output[k] -= m[i - 2] * output[j]; /* i >= 2 */
        }
    }

    /* back substitution */
    if( n > 3 )
        ts_arr_fill( output + lst, dim, 0.f );

    for( i = n - 2; i >= 1; i-- )
    {
        for( d = 0; d < dim; d++ )
        {
            k = i * dim + d;
            l = (i + 1) * dim + d;
            /* The following line is the reason why it's important to not fill
             * output with 0 if n = 3. On the other hand, if n > 3 subtracting
             * 0 is exactly what we want. */
            output[k] -= output[l];
            output[k] *= m[i - 1]; /* i >= 1 */
        }
    }

    if( n > 3 )
        memcpy( output + lst, points + lst, sof_c );

    /* we are done */
    free( m );
}


void ts_internal_relaxed_uniform_cubic_bspline( const tsReal* points,
        const size_t n,
        const size_t dim,
        tsBSpline* bspline,
        jmp_buf buf
        )
{
    const size_t order = 4;         /* The order of the spline to interpolate. */
    const tsReal as = 1.f / 6.f;    /* The value 'a sixth'. */
    const tsReal at = 1.f / 3.f;    /* The value 'a third'. */
    const tsReal tt = 2.f / 3.f;    /* The value 'two third'. */
    size_t sof_c;                   /* The size of a single control point. */
    const tsReal* b = points;       /* The array of the b values. */
    tsReal* s;                      /* The array of the s values. */
    size_t  i, d;                   /* Used in for loops */
    size_t  j, k, l;                /* Uses as temporary indices. */
    tsError e_;
    jmp_buf b_;

    /* input validation */
    if( dim == 0 )
        longjmp( buf, TS_DIM_ZERO );

    if( n <= 1 )
        longjmp( buf, TS_DEG_GE_NCTRLP );

    /* in the following n >= 2 applies */

    sof_c = dim * sizeof(tsReal);    /* dim > 0 implies sof_c > 0 */

    /* n >= 2 implies n-1 >= 1 implies (n-1)*4 >= 4 */
    ts_internal_bspline_new( (n - 1) * 4, dim, order - 1, TS_BEZIERS, bspline, buf );

    TRY( b_, e_ )
    s = (tsReal*) malloc( n * sof_c );

    if( s == NULL )
        longjmp( b_, TS_MALLOC );

    CATCH ts_bspline_free( bspline );

    longjmp( buf, e_ );
    ETRY
    /* set s_0 to b_0 and s_n = b_n */
    memcpy( s, b, sof_c );

    memcpy( s + (n - 1) * dim, b + (n - 1) * dim, sof_c );

    /* set s_i = 1/6*b_i + 2/3*b_{i-1} + 1/6*b_{i+1}*/
    for( i = 1; i < n - 1; i++ )
    {
        for( d = 0; d < dim; d++ )
        {
            j = (i - 1) * dim + d;
            k = i * dim + d;
            l = (i + 1) * dim + d;
            s[k] = as * b[j];
            s[k] += tt * b[k];
            s[k] += as * b[l];
        }
    }

    /* create beziers from b and s */
    for( i = 0; i < n - 1; i++ )
    {
        for( d = 0; d < dim; d++ )
        {
            j = i * dim + d;
            k = i * 4 * dim + d;
            l = (i + 1) * dim + d;
            bspline->ctrlp[k] = s[j];
            bspline->ctrlp[k + dim] = tt * b[j] + at * b[l];
            bspline->ctrlp[k + 2 * dim] = at * b[j] + tt * b[l];
            bspline->ctrlp[k + 3 * dim] = s[l];
        }
    }

    free( s );
}


void ts_internal_bspline_interpolate_cubic( const tsReal* points, const size_t n, const size_t dim,
        tsBSpline* bspline, jmp_buf buf
        )
{
    tsError e;
    jmp_buf b;
    tsReal* thomas = (tsReal*) malloc( n * dim * sizeof(tsReal) );

    if( thomas == NULL )
        longjmp( buf, TS_MALLOC );

    TRY( b, e )
    ts_internal_bspline_thomas_algorithm( points, n, dim, thomas, b );
    ts_internal_relaxed_uniform_cubic_bspline( thomas, n, dim, bspline, b );
    ETRY free( thomas );

    if( e < 0 )
        longjmp( buf, e );
}


void ts_internal_bspline_derive( const tsBSpline* original,
        tsBSpline* derivative, jmp_buf buf
        )
{
    const size_t sof_f = sizeof(tsReal);
    const size_t dim    = original->dim;
    const size_t deg    = original->deg;
    const size_t nc     = original->n_ctrlp;
    const size_t nk     = original->n_knots;
    tsReal* from_ctrlp  = original->ctrlp;
    tsReal* from_knots  = original->knots;
    tsReal* to_ctrlp    = NULL;
    tsReal* to_knots    = NULL;
    size_t  i, j, k;

    if( deg < 1 || nc < 2 )
        longjmp( buf, TS_UNDERIVABLE );

    if( original != derivative )
    {
        ts_internal_bspline_new( nc - 1, dim, deg - 1, TS_NONE, derivative, buf );
        to_ctrlp = derivative->ctrlp;
        to_knots = derivative->knots;
    }
    else
    {
        to_ctrlp = (tsReal*) malloc( ( (nc - 1) * dim + (nk - 2) ) * sof_f );

        if( to_ctrlp == NULL )
            longjmp( buf, TS_MALLOC );

        to_knots = to_ctrlp + (nc - 1) * dim;
    }

    for( i = 0; i < nc - 1; i++ )
    {
        for( j = 0; j < dim; j++ )
        {
            if( ts_fequals( from_knots[i + deg + 1], from_knots[i + 1] ) )
            {
                free( to_ctrlp );
                longjmp( buf, TS_UNDERIVABLE );
            }
            else
            {
                k = i * dim + j;
                to_ctrlp[k] = from_ctrlp[(i + 1) * dim + j] - from_ctrlp[k];
                to_ctrlp[k] *= deg;
                to_ctrlp[k] /= from_knots[i + deg + 1] - from_knots[i + 1];
            }
        }
    }

    memcpy( to_knots, from_knots + 1, (nk - 2) * sof_f );

    if( original == derivative )
    {
        /* free old memory */
        free( from_ctrlp );
        /* assign new values */
        derivative->deg = deg - 1;
        derivative->order = deg;
        derivative->n_ctrlp = nc - 1;
        derivative->n_knots = nk - 2;
        derivative->ctrlp   = to_ctrlp;
        derivative->knots   = to_knots;
    }
}


void ts_internal_bspline_buckle( const tsBSpline* bspline, const tsReal b,
        tsBSpline* buckled, jmp_buf buf
        )
{
    const tsReal b_hat = 1.f - b; /* The straightening factor. */
    const size_t dim    = bspline->dim;
    const size_t N      = bspline->n_ctrlp;
    const tsReal* p0    = bspline->ctrlp;       /* Pointer to first ctrlp. */
    const tsReal* pn_1 = p0 + (N - 1) * dim;    /* Pointer to the last ctrlp. */
    size_t i, d;                                /* Used in for loops. */

    ts_internal_bspline_copy( bspline, buckled, buf );

    for( i = 0; i < N; i++ )
    {
        for( d = 0; d < dim; d++ )
        {
            buckled->ctrlp[i * dim + d] =
                b * buckled->ctrlp[i * dim + d] +
                b_hat * ( p0[d] + ( (tsReal) i / (N - 1) ) * (pn_1[d] - p0[d]) );
        }
    }
}


void ts_internal_bspline_to_beziers( const tsBSpline* bspline,
        tsBSpline* beziers, jmp_buf buf
        )
{
    tsError e;
    jmp_buf b;
    const size_t deg = bspline->deg;
    const size_t order = bspline->order;
    tsBSpline tmp;
    int resize;     /* The number of control points to add/remove. */
    size_t k;       /* The index of the splitted knot value. */
    tsReal u_min;   /* The minimum of the knot values. */
    tsReal u_max;   /* The maximum of the knot values. */

    ts_internal_bspline_copy( bspline, &tmp, buf );

    TRY( b, e )
    /* fix first control point if necessary */
    u_min = tmp.knots[deg];

    if( !ts_fequals( tmp.knots[0], u_min ) )
    {
        ts_internal_bspline_split( &tmp, u_min, &tmp, &k, b );
        resize = (int) ( -1 * deg + (deg * 2 - k) );
        ts_internal_bspline_resize( &tmp, resize, 0, &tmp, b );
    }

    /* fix last control point if necessary */
    u_max = tmp.knots[tmp.n_knots - order];

    if( !ts_fequals( tmp.knots[tmp.n_knots - 1], u_max ) )
    {
        ts_internal_bspline_split( &tmp, u_max, &tmp, &k, b );
        resize = (int) ( -1 * deg + ( k - (tmp.n_knots - order) ) );
        ts_internal_bspline_resize( &tmp, resize, 1, &tmp, b );
    }

    k = order;

    while( k < tmp.n_knots - order )
    {
        ts_internal_bspline_split( &tmp, tmp.knots[k], &tmp, &k, b );
        k++;
    }

    if( bspline == beziers )
        ts_bspline_free( beziers );

    ts_bspline_move( &tmp, beziers );
    ETRY ts_bspline_free(& tmp );

    if( e < 0 )
        longjmp( buf, e );
}


void ts_internal_bspline_set_ctrlp( const tsBSpline* bspline, const tsReal* ctrlp,
        tsBSpline* result, jmp_buf buf
        )
{
    const size_t s = bspline->n_ctrlp * bspline->dim * sizeof(tsReal);

    ts_internal_bspline_copy( bspline, result, buf );
    memmove( result->ctrlp, ctrlp, s );
}


void ts_internal_bspline_set_knots( const tsBSpline* bspline, const tsReal* knots,
        tsBSpline* result, jmp_buf buf
        )
{
    const size_t s = bspline->n_knots * sizeof(tsReal);

    ts_internal_bspline_copy( bspline, result, buf );
    memmove( result->knots, knots, s );
}


/********************************************************
*                                                       *
* Interface implementation                              *
*                                                       *
********************************************************/
void ts_deboornet_default( tsDeBoorNet* deBoorNet )
{
    deBoorNet->u = 0.f;
    deBoorNet->k = 0;
    deBoorNet->s = 0;
    deBoorNet->h = 0;
    deBoorNet->dim = 0;
    deBoorNet->n_points = 0;
    deBoorNet->points   = NULL;
    deBoorNet->result   = NULL;
}


void ts_deboornet_move( tsDeBoorNet* from, tsDeBoorNet* to )
{
    if( from == to )
        return;

    to->u = from->u;
    to->k = from->k;
    to->s = from->s;
    to->h = from->h;
    to->dim = from->dim;
    to->n_points    = from->n_points;
    to->points  = from->points;
    to->result  = from->result;
    ts_deboornet_default( from );
}


void ts_deboornet_free( tsDeBoorNet* deBoorNet )
{
    if( deBoorNet->points != NULL )
        free( deBoorNet->points );    /* automatically frees the field result */

    ts_deboornet_default( deBoorNet );
}


void ts_bspline_default( tsBSpline* bspline )
{
    bspline->deg = 0;
    bspline->order = 0;
    bspline->dim = 0;
    bspline->n_ctrlp = 0;
    bspline->n_knots    = 0;
    bspline->ctrlp  = NULL;
    bspline->knots  = NULL;
}


void ts_bspline_free( tsBSpline* bspline )
{
    if( bspline->ctrlp != NULL )
        free( bspline->ctrlp );

    ts_bspline_default( bspline );
}


void ts_bspline_move( tsBSpline* from, tsBSpline* to )
{
    if( from == to )
        return;

    to->deg = from->deg;
    to->order = from->order;
    to->dim = from->dim;
    to->n_ctrlp = from->n_ctrlp;
    to->n_knots = from->n_knots;
    to->ctrlp   = from->ctrlp;
    to->knots   = from->knots;
    ts_bspline_default( from );
}


tsError ts_bspline_new( size_t n_ctrlp, size_t dim, size_t deg, tsBSplineType type,
        tsBSpline* bspline
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_new( n_ctrlp, dim, deg, type, bspline, buf );
    CATCH ts_bspline_default( bspline );

    ETRY
    return err;
}


tsError ts_bspline_interpolate_cubic( const tsReal* points, size_t n, size_t dim,
        tsBSpline* bspline
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_interpolate_cubic( points, n, dim, bspline, buf );
    CATCH ts_bspline_default( bspline );

    ETRY
    return err;
}


tsError ts_bspline_derive( const tsBSpline* original,
        tsBSpline* derivative
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_derive( original, derivative, buf );
    CATCH

    if( original != derivative )
        ts_bspline_default( derivative );

    ETRY
    return err;
}


tsError ts_deboornet_copy( const tsDeBoorNet* original,
        tsDeBoorNet* copy
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_deboornet_copy( original, copy, buf );
    CATCH

    if( original != copy )
        ts_deboornet_default( copy );

    ETRY
    return err;
}


tsError ts_bspline_copy( const tsBSpline* original,
        tsBSpline* copy
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_copy( original, copy, buf );
    CATCH

    if( original != copy )
        ts_bspline_default( copy );

    ETRY
    return err;
}


tsError ts_bspline_set_ctrlp( const tsBSpline* bspline, const tsReal* ctrlp,
        tsBSpline* result
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_set_ctrlp( bspline, ctrlp, result, buf );
    CATCH

    if( bspline != result )
        ts_bspline_default( result );

    ETRY
    return err;
}


tsError ts_bspline_set_knots( const tsBSpline* bspline, const tsReal* knots,
        tsBSpline* result
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_set_knots( bspline, knots, result, buf );
    CATCH

    if( bspline != result )
        ts_bspline_default( result );

    ETRY
    return err;
}


tsError ts_bspline_fill_knots( const tsBSpline* original,
        tsBSplineType type,
        tsReal min,
        tsReal max,
        tsBSpline* result
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_fill_knots( original, type, min, max, result, buf );
    CATCH

    if( original != result )
        ts_bspline_default( result );

    ETRY
    return err;
}


tsError ts_bspline_evaluate( const tsBSpline* bspline, tsReal u,
        tsDeBoorNet* deBoorNet
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_evaluate( bspline, u, deBoorNet, buf );
    CATCH ts_deboornet_default( deBoorNet );

    ETRY
    return err;
}


tsError ts_bspline_insert_knot( const tsBSpline* bspline, tsReal u, size_t n,
        tsBSpline* result, size_t* k
        )
{
    tsDeBoorNet net;
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_evaluate( bspline, u, &net, buf );
    ts_internal_bspline_insert_knot( bspline, &net, n, result, buf );
    *k = net.k + n;
    CATCH

    if( bspline != result )
        ts_bspline_default( result );

    *k = 0;
    ETRY ts_deboornet_free(& net );

    return err;
}


tsError ts_bspline_resize( const tsBSpline* bspline, int n, int back,
        tsBSpline* resized
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_resize( bspline, n, back, resized, buf );
    CATCH

    if( bspline != resized )
        ts_bspline_default( resized );

    ETRY
    return err;
}


tsError ts_bspline_split( const tsBSpline* bspline, tsReal u,
        tsBSpline* split, size_t* k
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_split( bspline, u, split, k, buf );
    CATCH

    if( bspline != split )
        ts_bspline_default( split );

    ETRY
    return err;
}


tsError ts_bspline_buckle( const tsBSpline* bspline, tsReal b,
        tsBSpline* buckled
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_buckle( bspline, b, buckled, buf );
    CATCH

    if( bspline != buckled )
        ts_bspline_default( buckled );

    ETRY
    return err;
}


tsError ts_bspline_to_beziers( const tsBSpline* bspline,
        tsBSpline* beziers
        )
{
    tsError err;
    jmp_buf buf;

    TRY( buf, err )
    ts_internal_bspline_to_beziers( bspline, beziers, buf );
    CATCH

    if( bspline != beziers )
        ts_bspline_default( beziers );

    ETRY
    return err;
}


int ts_fequals( tsReal x, tsReal y )
{
    if( fabs( x - y ) <= FLT_MAX_ABS_ERROR )
    {
        return 1;
    }
    else
    {
        const tsReal r = (tsReal) fabs( x ) > (tsReal) fabs( y ) ?
                         (tsReal) fabs( (x - y) / x ) : (tsReal) fabs( (x - y) / y );
        return r <= FLT_MAX_REL_ERROR;
    }
}


const char* ts_enum_str( tsError err )
{
    if( err == TS_MALLOC )
        return "malloc failed";
    else if( err == TS_DIM_ZERO )
        return "dim == 0";
    else if( err == TS_DEG_GE_NCTRLP )
        return "deg >= #ctrlp";
    else if( err == TS_U_UNDEFINED )
        return "spline is undefined at given u";
    else if( err == TS_MULTIPLICITY )
        return "s > order";
    else if( err == TS_KNOTS_DECR )
        return "decreasing knot vector";
    else if( err == TS_NUM_KNOTS )
        return "unexpected number of knots";
    else if( err == TS_UNDERIVABLE )
        return "spline is not derivable";

    return "unknown error";
}


tsError ts_str_enum( const char* str )
{
    if( !strcmp( str, ts_enum_str( TS_MALLOC ) ) )
        return TS_MALLOC;
    else if( !strcmp( str, ts_enum_str( TS_DIM_ZERO ) ) )
        return TS_DIM_ZERO;
    else if( !strcmp( str, ts_enum_str( TS_DEG_GE_NCTRLP ) ) )
        return TS_DEG_GE_NCTRLP;
    else if( !strcmp( str, ts_enum_str( TS_U_UNDEFINED ) ) )
        return TS_U_UNDEFINED;
    else if( !strcmp( str, ts_enum_str( TS_MULTIPLICITY ) ) )
        return TS_MULTIPLICITY;
    else if( !strcmp( str, ts_enum_str( TS_KNOTS_DECR ) ) )
        return TS_KNOTS_DECR;
    else if( !strcmp( str, ts_enum_str( TS_NUM_KNOTS ) ) )
        return TS_NUM_KNOTS;
    else if( !strcmp( str, ts_enum_str( TS_UNDERIVABLE ) ) )
        return TS_UNDERIVABLE;

    return TS_SUCCESS;
}


void ts_arr_fill( tsReal* arr, size_t num, tsReal val )
{
    size_t i;

    for( i = 0; i < num; i++ )
        arr[i] = val;
}


tsReal ts_ctrlp_dist2( const tsReal* x, const tsReal* y, size_t dim
        )
{
    tsReal sum = 0;
    size_t i;

    for( i = 0; i < dim; i++ )
        sum += (x[i] - y[i]) * (x[i] - y[i]);

    return (tsReal) sqrt( sum );
}