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bezier-utils.cpp
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#define __SP_BEZIER_UTILS_C__

/** \file
 * Bezier interpolation for inkscape drawing code.
 */
/*
 * Original code published in:
 *   An Algorithm for Automatically Fitting Digitized Curves
 *   by Philip J. Schneider
 *  "Graphics Gems", Academic Press, 1990
 *
 * Authors:
 *   Philip J. Schneider
 *   Lauris Kaplinski <lauris@kaplinski.com>
 *   Peter Moulder <pmoulder@mail.csse.monash.edu.au>
 *
 * Copyright (C) 1990 Philip J. Schneider
 * Copyright (C) 2001 Lauris Kaplinski
 * Copyright (C) 2001 Ximian, Inc.
 * Copyright (C) 2003,2004 Monash University
 *
 * This library is free software; you can redistribute it and/or
 * modify it either under the terms of the GNU Lesser General Public
 * License version 2.1 as published by the Free Software Foundation
 * (the "LGPL") or, at your option, under the terms of the Mozilla
 * Public License Version 1.1 (the "MPL"). If you do not alter this
 * notice, a recipient may use your version of this file under either
 * the MPL or the LGPL.
 *
 * You should have received a copy of the LGPL along with this library
 * in the file COPYING-LGPL-2.1; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
 * You should have received a copy of the MPL along with this library
 * in the file COPYING-MPL-1.1
 *
 * The contents of this file are subject to the Mozilla Public License
 * Version 1.1 (the "License"); you may not use this file except in
 * compliance with the License. You may obtain a copy of the License at
 * http://www.mozilla.org/MPL/
 *
 * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
 * OF ANY KIND, either express or implied. See the LGPL or the MPL for
 * the specific language governing rights and limitations.
 *
 */

#define SP_HUGE 1e5
#define noBEZIER_DEBUG

#ifdef HAVE_IEEEFP_H
# include <ieeefp.h>
#endif

#include <2geom/bezier-utils.h>

#include <2geom/isnan.h>
#include <assert.h>

namespace Geom{

typedef Point BezierCurve[];

/* Forward declarations */
static void generate_bezier(Point b[], Point const d[], double const u[], unsigned len,
                            Point const &tHat1, Point const &tHat2, double tolerance_sq);
static void estimate_lengths(Point bezier[],
                             Point const data[], double const u[], unsigned len,
                             Point const &tHat1, Point const &tHat2);
static void estimate_bi(Point b[4], unsigned ei,
                        Point const data[], double const u[], unsigned len);
static void reparameterize(Point const d[], unsigned len, double u[], BezierCurve const bezCurve);
static double NewtonRaphsonRootFind(BezierCurve const Q, Point const &P, double u);
static Point darray_center_tangent(Point const d[], unsigned center, unsigned length);
static Point darray_right_tangent(Point const d[], unsigned const len);
static unsigned copy_without_nans_or_adjacent_duplicates(Point const src[], unsigned src_len, Point dest[]);
static void chord_length_parameterize(Point const d[], double u[], unsigned len);
static double compute_max_error_ratio(Point const d[], double const u[], unsigned len,
                                      BezierCurve const bezCurve, double tolerance,
                                      unsigned *splitPoint);
static double compute_hook(Point const &a, Point const &b, double const u, BezierCurve const bezCurve,
                           double const tolerance);


static Point const unconstrained_tangent(0, 0);


/*
 *  B0, B1, B2, B3 : Bezier multipliers
 */

#define B0(u) ( ( 1.0 - u )  *  ( 1.0 - u )  *  ( 1.0 - u ) )
#define B1(u) ( 3 * u  *  ( 1.0 - u )  *  ( 1.0 - u ) )
#define B2(u) ( 3 * u * u  *  ( 1.0 - u ) )
#define B3(u) ( u * u * u )

#ifdef BEZIER_DEBUG
# define DOUBLE_ASSERT(x) assert( ( (x) > -SP_HUGE ) && ( (x) < SP_HUGE ) )
# define BEZIER_ASSERT(b) do { \
           DOUBLE_ASSERT((b)[0][X]); DOUBLE_ASSERT((b)[0][Y]);  \
           DOUBLE_ASSERT((b)[1][X]); DOUBLE_ASSERT((b)[1][Y]);  \
           DOUBLE_ASSERT((b)[2][X]); DOUBLE_ASSERT((b)[2][Y]);  \
           DOUBLE_ASSERT((b)[3][X]); DOUBLE_ASSERT((b)[3][Y]);  \
         } while(0)
#else
# define DOUBLE_ASSERT(x) do { } while(0)
# define BEZIER_ASSERT(b) do { } while(0)
#endif


/**
 * Fit a single-segment Bezier curve to a set of digitized points.
 *
 * \return Number of segments generated, or -1 on error.
 */
int
00116 bezier_fit_cubic(Point *bezier, Point const *data, int len, double error)
{
    return bezier_fit_cubic_r(bezier, data, len, error, 1);
}

/**
 * Fit a multi-segment Bezier curve to a set of digitized points, with
 * possible weedout of identical points and NaNs.
 *
 * \param max_beziers Maximum number of generated segments
 * \param Result array, must be large enough for n. segments * 4 elements.
 *
 * \return Number of segments generated, or -1 on error.
 */
int
00131 bezier_fit_cubic_r(Point bezier[], Point const data[], int const len, double const error, unsigned const max_beziers)
{
    if(bezier == NULL || 
       data == NULL || 
       len <= 0 || 
       max_beziers >= (1ul << (31 - 2 - 1 - 3))) 
        return -1;
    
    Point *uniqued_data = new Point[len];
    unsigned uniqued_len = copy_without_nans_or_adjacent_duplicates(data, len, uniqued_data);

    if ( uniqued_len < 2 ) {
        delete[] uniqued_data;
        return 0;
    }

    /* Call fit-cubic function with recursion. */
    int const ret = bezier_fit_cubic_full(bezier, NULL, uniqued_data, uniqued_len,
                                          unconstrained_tangent, unconstrained_tangent,
                                          error, max_beziers);
    delete[] uniqued_data;
    return ret;
}

/** 
 * Copy points from src to dest, filter out points containing NaN and
 * adjacent points with equal x and y.
 * \return length of dest
 */
static unsigned
00161 copy_without_nans_or_adjacent_duplicates(Point const src[], unsigned src_len, Point dest[])
{
    unsigned si = 0;
    for (;;) {
        if ( si == src_len ) {
            return 0;
        }
        if (!IS_NAN(src[si][X]) &&
            !IS_NAN(src[si][Y])) {
            dest[0] = Point(src[si]);
            ++si;
            break;
        }
        si++;
    }
    unsigned di = 0;
    for (; si < src_len; ++si) {
        Point const src_pt = Point(src[si]);
        if ( src_pt != dest[di]
             && !IS_NAN(src_pt[X])
             && !IS_NAN(src_pt[Y])) {
            dest[++di] = src_pt;
        }
    }
    unsigned dest_len = di + 1;
    assert( dest_len <= src_len );
    return dest_len;
}

/**
 * Fit a multi-segment Bezier curve to a set of digitized points, without
 * possible weedout of identical points and NaNs.
 * 
 * \pre data is uniqued, i.e. not exist i: data[i] == data[i + 1].
 * \param max_beziers Maximum number of generated segments
 * \param Result array, must be large enough for n. segments * 4 elements.
 */
int
00199 bezier_fit_cubic_full(Point bezier[], int split_points[],
                      Point const data[], int const len,
                      Point const &tHat1, Point const &tHat2,
                      double const error, unsigned const max_beziers)
{
    int const maxIterations = 4;   /* std::max times to try iterating */
    
    if(!(bezier != NULL) ||
       !(data != NULL) ||
       !(len > 0) ||
       !(max_beziers >= 1) ||
       !(error >= 0.0))
        return -1;

    if ( len < 2 ) return 0;

    if ( len == 2 ) {
        /* We have 2 points, which can be fitted trivially. */
        bezier[0] = data[0];
        bezier[3] = data[len - 1];
        double const dist = distance(bezier[0], bezier[3]) / 3.0;
        if (IS_NAN(dist)) {
            /* Numerical problem, fall back to straight line segment. */
            bezier[1] = bezier[0];
            bezier[2] = bezier[3];
        } else {
            bezier[1] = ( is_zero(tHat1)
                          ? ( 2 * bezier[0] + bezier[3] ) / 3.
                          : bezier[0] + dist * tHat1 );
            bezier[2] = ( is_zero(tHat2)
                          ? ( bezier[0] + 2 * bezier[3] ) / 3.
                          : bezier[3] + dist * tHat2 );
        }
        BEZIER_ASSERT(bezier);
        return 1;
    }

    /*  Parameterize points, and attempt to fit curve */
    unsigned splitPoint;   /* Point to split point set at. */
    bool is_corner;
    {
        double *u = new double[len];
        chord_length_parameterize(data, u, len);
        if ( u[len - 1] == 0.0 ) {
            /* Zero-length path: every point in data[] is the same.
             *
             * (Clients aren't allowed to pass such data; handling the case is defensive
             * programming.)
             */
            delete[] u;
            return 0;
        }

        generate_bezier(bezier, data, u, len, tHat1, tHat2, error);
        reparameterize(data, len, u, bezier);

        /* Find max deviation of points to fitted curve. */
        double const tolerance = sqrt(error + 1e-9);
        double maxErrorRatio = compute_max_error_ratio(data, u, len, bezier, tolerance, &splitPoint);

        if ( fabs(maxErrorRatio) <= 1.0 ) {
            BEZIER_ASSERT(bezier);
            delete[] u;
            return 1;
        }

        /* If error not too large, then try some reparameterization and iteration. */
        if ( 0.0 <= maxErrorRatio && maxErrorRatio <= 3.0 ) {
            for (int i = 0; i < maxIterations; i++) {
                generate_bezier(bezier, data, u, len, tHat1, tHat2, error);
                reparameterize(data, len, u, bezier);
                maxErrorRatio = compute_max_error_ratio(data, u, len, bezier, tolerance, &splitPoint);
                if ( fabs(maxErrorRatio) <= 1.0 ) {
                    BEZIER_ASSERT(bezier);
                    delete[] u;
                    return 1;
                }
            }
        }
        delete[] u;
        is_corner = (maxErrorRatio < 0);
    }

    if (is_corner) {
        assert(splitPoint < unsigned(len));
        if (splitPoint == 0) {
            if (is_zero(tHat1)) {
                /* Got spike even with unconstrained initial tangent. */
                ++splitPoint;
            } else {
                return bezier_fit_cubic_full(bezier, split_points, data, len, unconstrained_tangent, tHat2,
                                                error, max_beziers);
            }
        } else if (splitPoint == unsigned(len - 1)) {
            if (is_zero(tHat2)) {
                /* Got spike even with unconstrained final tangent. */
                --splitPoint;
            } else {
                return bezier_fit_cubic_full(bezier, split_points, data, len, tHat1, unconstrained_tangent,
                                                error, max_beziers);
            }
        }
    }

    if ( 1 < max_beziers ) {
        /*
         *  Fitting failed -- split at max error point and fit recursively
         */
        unsigned const rec_max_beziers1 = max_beziers - 1;

        Point recTHat2, recTHat1;
        if (is_corner) {
            if(!(0 < splitPoint && splitPoint < unsigned(len - 1)))
               return -1;
            recTHat1 = recTHat2 = unconstrained_tangent;
        } else {
            /* Unit tangent vector at splitPoint. */
            recTHat2 = darray_center_tangent(data, splitPoint, len);
            recTHat1 = -recTHat2;
        }
        int const nsegs1 = bezier_fit_cubic_full(bezier, split_points, data, splitPoint + 1,
                                                     tHat1, recTHat2, error, rec_max_beziers1);
        if ( nsegs1 < 0 ) {
#ifdef BEZIER_DEBUG
            g_print("fit_cubic[1]: recursive call failed\n");
#endif
            return -1;
        }
        assert( nsegs1 != 0 );
        if (split_points != NULL) {
            split_points[nsegs1 - 1] = splitPoint;
        }
        unsigned const rec_max_beziers2 = max_beziers - nsegs1;
        int const nsegs2 = bezier_fit_cubic_full(bezier + nsegs1*4,
                                                     ( split_points == NULL
                                                       ? NULL
                                                       : split_points + nsegs1 ),
                                                     data + splitPoint, len - splitPoint,
                                                     recTHat1, tHat2, error, rec_max_beziers2);
        if ( nsegs2 < 0 ) {
#ifdef BEZIER_DEBUG
            g_print("fit_cubic[2]: recursive call failed\n");
#endif
            return -1;
        }

#ifdef BEZIER_DEBUG
        g_print("fit_cubic: success[nsegs: %d+%d=%d] on max_beziers:%u\n",
                nsegs1, nsegs2, nsegs1 + nsegs2, max_beziers);
#endif
        return nsegs1 + nsegs2;
    } else {
        return -1;
    }
}


/**
 * Fill in \a bezier[] based on the given data and tangent requirements, using
 * a least-squares fit.
 *
 * Each of tHat1 and tHat2 should be either a zero vector or a unit vector.
 * If it is zero, then bezier[1 or 2] is estimated without constraint; otherwise,
 * it bezier[1 or 2] is placed in the specified direction from bezier[0 or 3].
 *
 * \param tolerance_sq Used only for an initial guess as to tangent directions
 *   when \a tHat1 or \a tHat2 is zero.
 */
static void
00368 generate_bezier(Point bezier[],
                Point const data[], double const u[], unsigned const len,
                Point const &tHat1, Point const &tHat2,
                double const tolerance_sq)
{
    bool const est1 = is_zero(tHat1);
    bool const est2 = is_zero(tHat2);
    Point est_tHat1( est1
                         ? darray_left_tangent(data, len, tolerance_sq)
                         : tHat1 );
    Point est_tHat2( est2
                         ? darray_right_tangent(data, len, tolerance_sq)
                         : tHat2 );
    estimate_lengths(bezier, data, u, len, est_tHat1, est_tHat2);
    /* We find that darray_right_tangent tends to produce better results
       for our current freehand tool than full estimation. */
    if (est1) {
        estimate_bi(bezier, 1, data, u, len);
        if (bezier[1] != bezier[0]) {
            est_tHat1 = unit_vector(bezier[1] - bezier[0]);
        }
        estimate_lengths(bezier, data, u, len, est_tHat1, est_tHat2);
    }
}


static void
00395 estimate_lengths(Point bezier[],
                 Point const data[], double const uPrime[], unsigned const len,
                 Point const &tHat1, Point const &tHat2)
{
    double C[2][2];   /* Matrix C. */
    double X[2];      /* Matrix X. */

    /* Create the C and X matrices. */
    C[0][0] = 0.0;
    C[0][1] = 0.0;
    C[1][0] = 0.0;
    C[1][1] = 0.0;
    X[0]    = 0.0;
    X[1]    = 0.0;

    /* First and last control points of the Bezier curve are positioned exactly at the first and
       last data points. */
    bezier[0] = data[0];
    bezier[3] = data[len - 1];

    for (unsigned i = 0; i < len; i++) {
        /* Bezier control point coefficients. */
        double const b0 = B0(uPrime[i]);
        double const b1 = B1(uPrime[i]);
        double const b2 = B2(uPrime[i]);
        double const b3 = B3(uPrime[i]);

        /* rhs for eqn */
        Point const a1 = b1 * tHat1;
        Point const a2 = b2 * tHat2;

        C[0][0] += dot(a1, a1);
        C[0][1] += dot(a1, a2);
        C[1][0] = C[0][1];
        C[1][1] += dot(a2, a2);

        /* Additional offset to the data point from the predicted point if we were to set bezier[1]
           to bezier[0] and bezier[2] to bezier[3]. */
        Point const shortfall
            = ( data[i]
                - ( ( b0 + b1 ) * bezier[0] )
                - ( ( b2 + b3 ) * bezier[3] ) );
        X[0] += dot(a1, shortfall);
        X[1] += dot(a2, shortfall);
    }

    /* We've constructed a pair of equations in the form of a matrix product C * alpha = X.
       Now solve for alpha. */
    double alpha_l, alpha_r;

    /* Compute the determinants of C and X. */
    double const det_C0_C1 = C[0][0] * C[1][1] - C[1][0] * C[0][1];
    if ( det_C0_C1 != 0 ) {
        /* Apparently Kramer's rule. */
        double const det_C0_X  = C[0][0] * X[1]    - C[0][1] * X[0];
        double const det_X_C1  = X[0]    * C[1][1] - X[1]    * C[0][1];
        alpha_l = det_X_C1 / det_C0_C1;
        alpha_r = det_C0_X / det_C0_C1;
    } else {
        /* The matrix is under-determined.  Try requiring alpha_l == alpha_r.
         *
         * One way of implementing the constraint alpha_l == alpha_r is to treat them as the same
         * variable in the equations.  We can do this by adding the columns of C to form a single
         * column, to be multiplied by alpha to give the column vector X.
         *
         * We try each row in turn.
         */
        double const c0 = C[0][0] + C[0][1];
        if (c0 != 0) {
            alpha_l = alpha_r = X[0] / c0;
        } else {
            double const c1 = C[1][0] + C[1][1];
            if (c1 != 0) {
                alpha_l = alpha_r = X[1] / c1;
            } else {
                /* Let the below code handle this. */
                alpha_l = alpha_r = 0.;
            }
        }
    }

    /* If alpha negative, use the Wu/Barsky heuristic (see text).  (If alpha is 0, you get
       coincident control points that lead to divide by zero in any subsequent
       NewtonRaphsonRootFind() call.) */
    /// \todo Check whether this special-casing is necessary now that 
    /// NewtonRaphsonRootFind handles non-positive denominator.
    if ( alpha_l < 1.0e-6 ||
         alpha_r < 1.0e-6   )
    {
        alpha_l = alpha_r = distance(data[0], data[len-1]) / 3.0;
    }

    /* Control points 1 and 2 are positioned an alpha distance out on the tangent vectors, left and
       right, respectively. */
    bezier[1] = alpha_l * tHat1 + bezier[0];
    bezier[2] = alpha_r * tHat2 + bezier[3];

    return;
}

static double lensq(Point const p) {
    return dot(p, p);
}

static void
estimate_bi(Point bezier[4], unsigned const ei,
            Point const data[], double const u[], unsigned const len)
{
    if(!(1 <= ei && ei <= 2))
        return;
    unsigned const oi = 3 - ei;
    double num[2] = {0., 0.};
    double den = 0.;
    for (unsigned i = 0; i < len; ++i) {
        double const ui = u[i];
        double const b[4] = {
            B0(ui),
            B1(ui),
            B2(ui),
            B3(ui)
        };

        for (unsigned d = 0; d < 2; ++d) {
            num[d] += b[ei] * (b[0]  * bezier[0][d] +
                               b[oi] * bezier[oi][d] +
                               b[3]  * bezier[3][d] +
                               - data[i][d]);
        }
        den -= b[ei] * b[ei];
    }

    if (den != 0.) {
        for (unsigned d = 0; d < 2; ++d) {
            bezier[ei][d] = num[d] / den;
        }
    } else {
        bezier[ei] = ( oi * bezier[0] + ei * bezier[3] ) / 3.;
    }
}

/**
 * Given set of points and their parameterization, try to find a better assignment of parameter
 * values for the points.
 *
 *  \param d  Array of digitized points.
 *  \param u  Current parameter values.
 *  \param bezCurve  Current fitted curve.
 *  \param len  Number of values in both d and u arrays.
 *              Also the size of the array that is allocated for return.
 */
static void
00546 reparameterize(Point const d[],
               unsigned const len,
               double u[],
               BezierCurve const bezCurve)
{
    assert( 2 <= len );

    unsigned const last = len - 1;
    assert( bezCurve[0] == d[0] );
    assert( bezCurve[3] == d[last] );
    assert( u[0] == 0.0 );
    assert( u[last] == 1.0 );
    /* Otherwise, consider including 0 and last in the below loop. */

    for (unsigned i = 1; i < last; i++) {
        u[i] = NewtonRaphsonRootFind(bezCurve, d[i], u[i]);
    }
}

/**
 *  Use Newton-Raphson iteration to find better root.
 *  
 *  \param Q  Current fitted curve
 *  \param P  Digitized point
 *  \param u  Parameter value for "P"
 *  
 *  \return Improved u
 */
static double
00575 NewtonRaphsonRootFind(BezierCurve const Q, Point const &P, double const u)
{
    assert( 0.0 <= u );
    assert( u <= 1.0 );

    /* Generate control vertices for Q'. */
    Point Q1[3];
    for (unsigned i = 0; i < 3; i++) {
        Q1[i] = 3.0 * ( Q[i+1] - Q[i] );
    }

    /* Generate control vertices for Q''. */
    Point Q2[2];
    for (unsigned i = 0; i < 2; i++) {
        Q2[i] = 2.0 * ( Q1[i+1] - Q1[i] );
    }

    /* Compute Q(u), Q'(u) and Q''(u). */
    Point const Q_u  = bezier_pt(3, Q, u);
    Point const Q1_u = bezier_pt(2, Q1, u);
    Point const Q2_u = bezier_pt(1, Q2, u);

    /* Compute f(u)/f'(u), where f is the derivative wrt u of distsq(u) = 0.5 * the square of the
       distance from P to Q(u).  Here we're using Newton-Raphson to find a stationary point in the
       distsq(u), hopefully corresponding to a local minimum in distsq (and hence a local minimum
       distance from P to Q(u)). */
    Point const diff = Q_u - P;
    double numerator = dot(diff, Q1_u);
    double denominator = dot(Q1_u, Q1_u) + dot(diff, Q2_u);

    double improved_u;
    if ( denominator > 0. ) {
        /* One iteration of Newton-Raphson:
           improved_u = u - f(u)/f'(u) */
        improved_u = u - ( numerator / denominator );
    } else {
        /* Using Newton-Raphson would move in the wrong direction (towards a local maximum rather
           than local minimum), so we move an arbitrary amount in the right direction. */
        if ( numerator > 0. ) {
            improved_u = u * .98 - .01;
        } else if ( numerator < 0. ) {
            /* Deliberately asymmetrical, to reduce the chance of cycling. */
            improved_u = .031 + u * .98;
        } else {
            improved_u = u;
        }
    }

    if (!IS_FINITE(improved_u)) {
        improved_u = u;
    } else if ( improved_u < 0.0 ) {
        improved_u = 0.0;
    } else if ( improved_u > 1.0 ) {
        improved_u = 1.0;
    }

    /* Ensure that improved_u isn't actually worse. */
    {
        double const diff_lensq = lensq(diff);
        for (double proportion = .125; ; proportion += .125) {
            if ( lensq( bezier_pt(3, Q, improved_u) - P ) > diff_lensq ) {
                if ( proportion > 1.0 ) {
                    //g_warning("found proportion %g", proportion);
                    improved_u = u;
                    break;
                }
                improved_u = ( ( 1 - proportion ) * improved_u  +
                               proportion         * u            );
            } else {
                break;
            }
        }
    }

    DOUBLE_ASSERT(improved_u);
    return improved_u;
}

/** 
 * Evaluate a Bezier curve at parameter value \a t.
 * 
 * \param degree The degree of the Bezier curve: 3 for cubic, 2 for quadratic etc. Must be less
 *    than 4.
 * \param V The control points for the Bezier curve.  Must have (\a degree+1)
 *    elements.
 * \param t The "parameter" value, specifying whereabouts along the curve to
 *    evaluate.  Typically in the range [0.0, 1.0].
 *
 * Let s = 1 - t.
 * BezierII(1, V) gives (s, t) * V, i.e. t of the way
 * from V[0] to V[1].
 * BezierII(2, V) gives (s**2, 2*s*t, t**2) * V.
 * BezierII(3, V) gives (s**3, 3 s**2 t, 3s t**2, t**3) * V.
 *
 * The derivative of BezierII(i, V) with respect to t
 * is i * BezierII(i-1, V'), where for all j, V'[j] =
 * V[j + 1] - V[j].
 */
Point
00674 bezier_pt(unsigned const degree, Point const V[], double const t)
{
    /** Pascal's triangle. */
    static int const pascal[4][4] = {{1},
                                     {1, 1},
                                     {1, 2, 1},
                                     {1, 3, 3, 1}};
    assert( degree < 4);
    double const s = 1.0 - t;

    /* Calculate powers of t and s. */
    double spow[4];
    double tpow[4];
    spow[0] = 1.0; spow[1] = s;
    tpow[0] = 1.0; tpow[1] = t;
    for (unsigned i = 1; i < degree; ++i) {
        spow[i + 1] = spow[i] * s;
        tpow[i + 1] = tpow[i] * t;
    }

    Point ret = spow[degree] * V[0];
    for (unsigned i = 1; i <= degree; ++i) {
        ret += pascal[degree][i] * spow[degree - i] * tpow[i] * V[i];
    }
    return ret;
}

/*
 * ComputeLeftTangent, ComputeRightTangent, ComputeCenterTangent :
 * Approximate unit tangents at endpoints and "center" of digitized curve
 */

/** 
 * Estimate the (forward) tangent at point d[first + 0.5].
 *
 * Unlike the center and right versions, this calculates the tangent in 
 * the way one might expect, i.e., wrt increasing index into d.
 * \pre (2 <= len) and (d[0] != d[1]).
 **/
Point
00714 darray_left_tangent(Point const d[], unsigned const len)
{
    assert( len >= 2 );
    assert( d[0] != d[1] );
    return unit_vector( d[1] - d[0] );
}

/** 
 * Estimates the (backward) tangent at d[last - 0.5].
 *
 * \note The tangent is "backwards", i.e. it is with respect to 
 * decreasing index rather than increasing index.
 *
 * \pre 2 <= len.
 * \pre d[len - 1] != d[len - 2].
 * \pre all[p in d] in_svg_plane(p).
 */
static Point
00732 darray_right_tangent(Point const d[], unsigned const len)
{
    assert( 2 <= len );
    unsigned const last = len - 1;
    unsigned const prev = last - 1;
    assert( d[last] != d[prev] );
    return unit_vector( d[prev] - d[last] );
}

/** 
 * Estimate the (forward) tangent at point d[0].
 *
 * Unlike the center and right versions, this calculates the tangent in 
 * the way one might expect, i.e., wrt increasing index into d.
 *
 * \pre 2 <= len.
 * \pre d[0] != d[1].
 * \pre all[p in d] in_svg_plane(p).
 * \post is_unit_vector(ret).
 **/
Point
00753 darray_left_tangent(Point const d[], unsigned const len, double const tolerance_sq)
{
    assert( 2 <= len );
    assert( 0 <= tolerance_sq );
    for (unsigned i = 1;;) {
        Point const pi(d[i]);
        Point const t(pi - d[0]);
        double const distsq = dot(t, t);
        if ( tolerance_sq < distsq ) {
            return unit_vector(t);
        }
        ++i;
        if (i == len) {
            return ( distsq == 0
                     ? darray_left_tangent(d, len)
                     : unit_vector(t) );
        }
    }
}

/** 
 * Estimates the (backward) tangent at d[last].
 *
 * \note The tangent is "backwards", i.e. it is with respect to 
 * decreasing index rather than increasing index.
 *
 * \pre 2 <= len.
 * \pre d[len - 1] != d[len - 2].
 * \pre all[p in d] in_svg_plane(p).
 */
Point
00784 darray_right_tangent(Point const d[], unsigned const len, double const tolerance_sq)
{
    assert( 2 <= len );
    assert( 0 <= tolerance_sq );
    unsigned const last = len - 1;
    for (unsigned i = last - 1;; i--) {
        Point const pi(d[i]);
        Point const t(pi - d[last]);
        double const distsq = dot(t, t);
        if ( tolerance_sq < distsq ) {
            return unit_vector(t);
        }
        if (i == 0) {
            return ( distsq == 0
                     ? darray_right_tangent(d, len)
                     : unit_vector(t) );
        }
    }
}

/** 
 * Estimates the (backward) tangent at d[center], by averaging the two 
 * segments connected to d[center] (and then normalizing the result).
 *
 * \note The tangent is "backwards", i.e. it is with respect to 
 * decreasing index rather than increasing index.
 *
 * \pre (0 < center < len - 1) and d is uniqued (at least in 
 * the immediate vicinity of \a center).
 */
static Point
00815 darray_center_tangent(Point const d[],
                         unsigned const center,
                         unsigned const len)
{
    assert( center != 0 );
    assert( center < len - 1 );

    Point ret;
    if ( d[center + 1] == d[center - 1] ) {
        /* Rotate 90 degrees in an arbitrary direction. */
        Point const diff = d[center] - d[center - 1];
        ret = rot90(diff);
    } else {
        ret = d[center - 1] - d[center + 1];
    }
    ret.normalize();
    return ret;
}


/**
 *  Assign parameter values to digitized points using relative distances between points.
 *
 *  \pre Parameter array u must have space for \a len items.
 */
static void
00841 chord_length_parameterize(Point const d[], double u[], unsigned const len)
{
    if(!( 2 <= len ))
        return;

    /* First let u[i] equal the distance travelled along the path from d[0] to d[i]. */
    u[0] = 0.0;
    for (unsigned i = 1; i < len; i++) {
        double const dist = distance(d[i], d[i-1]);
        u[i] = u[i-1] + dist;
    }

    /* Then scale to [0.0 .. 1.0]. */
    double tot_len = u[len - 1];
    if(!( tot_len != 0 ))
        return;
    if (IS_FINITE(tot_len)) {
        for (unsigned i = 1; i < len; ++i) {
            u[i] /= tot_len;
        }
    } else {
        /* We could do better, but this probably never happens anyway. */
        for (unsigned i = 1; i < len; ++i) {
            u[i] = i / (double) ( len - 1 );
        }
    }

    /** \todo
     * It's been reported that u[len - 1] can differ from 1.0 on some 
     * systems (amd64), despite it having been calculated as x / x where x 
     * is isFinite and non-zero.
     */
    if (u[len - 1] != 1) {
        double const diff = u[len - 1] - 1;
        if (fabs(diff) > 1e-13) {
            assert(0); // No warnings in 2geom
            //g_warning("u[len - 1] = %19g (= 1 + %19g), expecting exactly 1",
            //          u[len - 1], diff);
        }
        u[len - 1] = 1;
    }

#ifdef BEZIER_DEBUG
    assert( u[0] == 0.0 && u[len - 1] == 1.0 );
    for (unsigned i = 1; i < len; i++) {
        assert( u[i] >= u[i-1] );
    }
#endif
}




/**
 * Find the maximum squared distance of digitized points to fitted curve, and (if this maximum
 * error is non-zero) set \a *splitPoint to the corresponding index.
 *
 * \pre 2 <= len.
 * \pre u[0] == 0.
 * \pre u[len - 1] == 1.0.
 * \post ((ret == 0.0)
 *        || ((*splitPoint < len - 1)
 *            \&\& (*splitPoint != 0 || ret < 0.0))).
 */
static double
00906 compute_max_error_ratio(Point const d[], double const u[], unsigned const len,
                        BezierCurve const bezCurve, double const tolerance,
                        unsigned *const splitPoint)
{
    assert( 2 <= len );
    unsigned const last = len - 1;
    assert( bezCurve[0] == d[0] );
    assert( bezCurve[3] == d[last] );
    assert( u[0] == 0.0 );
    assert( u[last] == 1.0 );
    /* I.e. assert that the error for the first & last points is zero.
     * Otherwise we should include those points in the below loop.
     * The assertion is also necessary to ensure 0 < splitPoint < last.
     */

    double maxDistsq = 0.0; /* Maximum error */
    double max_hook_ratio = 0.0;
    unsigned snap_end = 0;
    Point prev = bezCurve[0];
    for (unsigned i = 1; i <= last; i++) {
        Point const curr = bezier_pt(3, bezCurve, u[i]);
        double const distsq = lensq( curr - d[i] );
        if ( distsq > maxDistsq ) {
            maxDistsq = distsq;
            *splitPoint = i;
        }
        double const hook_ratio = compute_hook(prev, curr, .5 * (u[i - 1] + u[i]), bezCurve, tolerance);
        if (max_hook_ratio < hook_ratio) {
            max_hook_ratio = hook_ratio;
            snap_end = i;
        }
        prev = curr;
    }

    double const dist_ratio = sqrt(maxDistsq) / tolerance;
    double ret;
    if (max_hook_ratio <= dist_ratio) {
        ret = dist_ratio;
    } else {
        assert(0 < snap_end);
        ret = -max_hook_ratio;
        *splitPoint = snap_end - 1;
    }
    assert( ret == 0.0
              || ( ( *splitPoint < last )
                   && ( *splitPoint != 0 || ret < 0. ) ) );
    return ret;
}

/** 
 * Whereas compute_max_error_ratio() checks for itself that each data point 
 * is near some point on the curve, this function checks that each point on 
 * the curve is near some data point (or near some point on the polyline 
 * defined by the data points, or something like that: we allow for a
 * "reasonable curviness" from such a polyline).  "Reasonable curviness" 
 * means we draw a circle centred at the midpoint of a..b, of radius 
 * proportional to the length |a - b|, and require that each point on the 
 * segment of bezCurve between the parameters of a and b be within that circle.
 * If any point P on the bezCurve segment is outside of that allowable 
 * region (circle), then we return some metric that increases with the 
 * distance from P to the circle.
 *
 *  Given that this is a fairly arbitrary criterion for finding appropriate 
 *  places for sharp corners, we test only one point on bezCurve, namely 
 *  the point on bezCurve with parameter halfway between our estimated 
 *  parameters for a and b.  (Alternatives are taking the farthest of a
 *  few parameters between those of a and b, or even using a variant of 
 *  NewtonRaphsonFindRoot() for finding the maximum rather than minimum 
 *  distance.)
 */
static double
00977 compute_hook(Point const &a, Point const &b, double const u, BezierCurve const bezCurve,
             double const tolerance)
{
    Point const P = bezier_pt(3, bezCurve, u);
    double const dist = distance((a+b)*.5, P);
    if (dist < tolerance) {
        return 0;
    }
    double const allowed = distance(a, b) + tolerance;
    return dist / allowed;
    /** \todo 
     * effic: Hooks are very rare.  We could start by comparing 
     * distsq, only resorting to the more expensive L2 in cases of 
     * uncertainty.
     */
}

}

/*
  Local Variables:
  mode:c++
  c-file-style:"stroustrup"
  c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
  indent-tabs-mode:nil
  fill-column:99
  End:
*/
// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :

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