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diff --git a/src/2geom/bezier-utils.cpp b/src/2geom/bezier-utils.cpp new file mode 100644 index 0000000..181b5b3 --- /dev/null +++ b/src/2geom/bezier-utils.cpp @@ -0,0 +1,997 @@ +/* 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/math-utils.h> +#include <assert.h> + +namespace Geom { + +/* 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[], Point const bezCurve[]); +static double NewtonRaphsonRootFind(Point 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, + Point const bezCurve[], double tolerance, + unsigned *splitPoint); +static double compute_hook(Point const &a, Point const &b, double const u, Point 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 +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 +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 +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 (!std::isnan(src[si][X]) && + !std::isnan(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] + && !std::isnan(src_pt[X]) + && !std::isnan(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 +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) +{ + 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 (std::isnan(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 ) { + int const maxIterations = 4; /* std::max times to try iterating */ + 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 +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 +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 +reparameterize(Point const d[], + unsigned const len, + double u[], + Point 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 +NewtonRaphsonRootFind(Point 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 (!std::isfinite(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 +bezier_pt(unsigned const degree, Point const V[], double const t) +{ + /** Pascal's triangle. */ + static int const pascal[4][4] = {{1, 0, 0, 0}, + {1, 1, 0, 0}, + {1, 2, 1, 0}, + {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 +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 +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 +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 +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 +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 +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 (std::isfinite(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 +compute_max_error_ratio(Point const d[], double const u[], unsigned const len, + Point 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 +compute_hook(Point const &a, Point const &b, double const u, Point 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:fileencoding=utf-8:textwidth=99 : |