X-Git-Url: https://git.donarmstrong.com/?a=blobdiff_plain;f=lily%2Fbezier.cc;h=7d3ad6af14fff2d8d2a24267f6fc9e9b729dff6b;hb=0b544cfb7332615ef809b71b57ab656741311ae1;hp=de47a3b0de205e6e17c4243d4c2fef1372d46fcf;hpb=92ee5a5e2dfa180b153666f560ea75c3dcb96290;p=lilypond.git diff --git a/lily/bezier.cc b/lily/bezier.cc index de47a3b0de..7d3ad6af14 100644 --- a/lily/bezier.cc +++ b/lily/bezier.cc @@ -1,124 +1,191 @@ /* - bezier.cc -- implement Bezier and Bezier_bow + This file is part of LilyPond, the GNU music typesetter. - source file of the GNU LilyPond music typesetter + Copyright (C) 1998--2014 Jan Nieuwenhuizen - (c) 1998--2005 Jan Nieuwenhuizen -*/ + LilyPond is free software: you can redistribute it and/or modify + it under the terms of the GNU General Public License as published by + the Free Software Foundation, either version 3 of the License, or + (at your option) any later version. + + LilyPond is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU General Public License for more details. -#include + You should have received a copy of the GNU General Public License + along with LilyPond. If not, see . +*/ #include "bezier.hh" #include "warn.hh" #include "libc-extension.hh" -Real -binomial_coefficient (Real over , int under) +Real binomial_coefficient_3[] += { - Real x = 1.0; - - while (under) - { - x *= over / Real (under); - - over -= 1.0; - under --; - } - return x; -} + 1, 3, 3, 1 +}; void -scale (Array* array, Real x , Real y) +scale (vector *array, Real x, Real y) { - for (int i = 0; i < array->size (); i++) + for (vsize i = 0; i < array->size (); i++) { - (*array)[i][X_AXIS] = x* (*array)[i][X_AXIS]; - (*array)[i][Y_AXIS] = y* (*array)[i][Y_AXIS]; + (*array)[i][X_AXIS] = x * (*array)[i][X_AXIS]; + (*array)[i][Y_AXIS] = y * (*array)[i][Y_AXIS]; } } void -rotate (Array* array, Real phi) +rotate (vector *array, Real phi) { Offset rot (complex_exp (Offset (0, phi))); - for (int i = 0; i < array->size (); i++) + for (vsize i = 0; i < array->size (); i++) (*array)[i] = complex_multiply (rot, (*array)[i]); } void -translate (Array* array, Offset o) +translate (vector *array, Offset o) { - for (int i = 0; i < array->size (); i++) + for (vsize i = 0; i < array->size (); i++) (*array)[i] += o; } /* - Formula of the bezier 3-spline sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j) t^j A is the axis of X coordinate. - */ +*/ Real -Bezier::get_other_coordinate (Axis a, Real x) const +Bezier::get_other_coordinate (Axis a, Real x) const { - Axis other = Axis ((a +1)%NO_AXES); - Array ts = solve_point (a, x); + Axis other = Axis ((a + 1) % NO_AXES); + vector ts = solve_point (a, x); if (ts.size () == 0) { - programming_error ("No solution found for Bezier intersection."); + programming_error ("no solution found for Bezier intersection"); return 0.0; } - - Offset c = curve_point (ts[0]); +#ifdef PARANOID + Offset c = curve_point (ts[0]); if (fabs (c[a] - x) > 1e-8) - programming_error ("Bezier intersection not correct?"); - - return c[other]; + programming_error ("bezier intersection not correct?"); +#endif + + return curve_coordinate (ts[0], other); } +vector +Bezier::get_other_coordinates (Axis a, Real x) const +{ + Axis other = other_axis (a); + vector ts = solve_point (a, x); + vector sols; + for (vsize i = 0; i < ts.size (); i++) + sols.push_back (curve_coordinate (ts[i], other)); + return sols; +} + +Real +Bezier::curve_coordinate (Real t, Axis a) const +{ + Real tj = 1; + Real one_min_tj[4]; + one_min_tj[0] = 1; + for (int i = 1; i < 4; i++) + one_min_tj[i] = one_min_tj[i - 1] * (1 - t); + + Real r = 0.0; + for (int j = 0; j < 4; j++) + { + r += control_[j][a] * binomial_coefficient_3[j] + * tj * one_min_tj[3 - j]; + + tj *= t; + } + + return r; +} Offset Bezier::curve_point (Real t) const { Real tj = 1; - Real one_min_tj = (1-t)* (1-t)* (1-t); + Real one_min_tj[4]; + one_min_tj[0] = 1; + for (int i = 1; i < 4; i++) + one_min_tj[i] = one_min_tj[i - 1] * (1 - t); Offset o; - for (int j = 0 ; j < 4; j++) + for (int j = 0; j < 4; j++) { - o += control_[j] * binomial_coefficient (3, j) - * pow (t, j) * pow (1-t, 3-j); + o += control_[j] * binomial_coefficient_3[j] + * tj * one_min_tj[3 - j]; tj *= t; - if (1-t) - one_min_tj /= (1-t); } #ifdef PARANOID - assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t))< 1e-8); - assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t))< 1e-8); + assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t)) < 1e-8); + assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t)) < 1e-8); #endif - + return o; } +Real +Bezier::slope_at_point (Real t) const +{ + Offset second_order[3]; + Offset third_order[2]; + + for (vsize i = 0; i < 3; i++) + second_order[i] = ((control_[i + 1] - control_[i]) * t) + control_[i]; + + for (vsize i = 0; i < 2; i++) + third_order[i] = ((second_order[i + 1] - second_order[i]) * t) + second_order[i]; + + if (third_order[1][X_AXIS] - third_order[0][X_AXIS] == 0) + return infinity_f; + + return (third_order[1][Y_AXIS] - third_order[0][Y_AXIS]) / (third_order[1][X_AXIS] - third_order[0][X_AXIS]); +} + +/* + Cache binom (3, j) t^j (1-t)^{3-j} +*/ +struct Polynomial_cache +{ + Polynomial terms_[4]; + Polynomial_cache () + { + for (int j = 0; j <= 3; j++) + terms_[j] + = binomial_coefficient_3[j] + * Polynomial::power (j, Polynomial (0, 1)) + * Polynomial::power (3 - j, Polynomial (1, -1)); + } +}; + +static Polynomial_cache poly_cache; Polynomial Bezier::polynomial (Axis a) const { Polynomial p (0.0); + Polynomial q; for (int j = 0; j <= 3; j++) { - p += - (control_[j][a] * binomial_coefficient (3, j)) - * Polynomial::power (j, Polynomial (0, 1)) - * Polynomial::power (3 - j, Polynomial (1, -1)); + q = poly_cache.terms_[j]; + q *= control_[j][a]; + p += q; } return p; @@ -126,60 +193,106 @@ Bezier::polynomial (Axis a) const /** Remove all numbers outside [0, 1] from SOL - */ -Array -filter_solutions (Array sol) +*/ +vector +filter_solutions (vector sol) { - for (int i = sol.size (); i--;) - if (sol[i] < 0 || sol[i] >1) - sol.del (i); + for (vsize i = sol.size (); i--;) + if (sol[i] < 0 || sol[i] > 1) + sol.erase (sol.begin () + i); return sol; } /** find t such that derivative is proportional to DERIV - */ -Array +*/ +vector Bezier::solve_derivative (Offset deriv) const { Polynomial xp = polynomial (X_AXIS); Polynomial yp = polynomial (Y_AXIS); xp.differentiate (); yp.differentiate (); - + Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS]; return filter_solutions (combine.solve ()); } - /* Find t such that curve_point (t)[AX] == COORDINATE */ -Array +vector Bezier::solve_point (Axis ax, Real coordinate) const { Polynomial p (polynomial (ax)); p.coefs_[0] -= coordinate; - - Array sol (p.solve ()); + + vector sol (p.solve ()); return filter_solutions (sol); } +/** + For the portion of the curve between L and R along axis AX, + return the bounding box limit in direction D along the cross axis to AX. + If there is no portion between L and R, return 0.0 and report error. +*/ +Real +Bezier::minmax (Axis ax, Real l, Real r, Direction d) const +{ + Axis bx = other_axis (ax); + + // The curve could hit its bounding box limit along BX at: + // points where the curve is parallel to AX, + Offset vec (0.0, 0.0); + vec[ax] = 1.0; + vector sols (solve_derivative (vec)); + // or endpoints of the curve, + sols.push_back (0.999); + sols.push_back (0.001); + // (using points just inside the ends, so that an endpoint is evaulated + // if it falls within rounding error of L or R and the curve lies inside) + + Interval iv; + for (vsize i = sols.size (); i--;) + { + Offset p (curve_point (sols[i])); + if (p[ax] >= l && p[ax] <= r) + iv.add_point (p[bx]); + } + + // or intersections of the curve with the bounding lines at L and R. + Interval lr (l, r); + for (LEFT_and_RIGHT (dir)) + { + vector v = get_other_coordinates (ax, lr[dir]); + for (vsize i = v.size (); i--;) + iv.add_point (v[i]); + } + + if (iv.is_empty ()) + { + programming_error ("Bezier curve does not cross region of concern"); + return 0.0; + } + + return iv.at (d); +} + /** Compute the bounding box dimensions in direction of A. - */ +*/ Interval Bezier::extent (Axis a) const { - int o = (a+1)%NO_AXES; + int o = (a + 1) % NO_AXES; Offset d; - d[Axis (o)] =1.0; + d[Axis (o)] = 1.0; Interval iv; - Array sols (solve_derivative (d)); - sols.push (1.0); - sols.push (0.0); - for (int i = sols.size (); i--;) + vector sols (solve_derivative (d)); + sols.push_back (1.0); + sols.push_back (0.0); + for (vsize i = sols.size (); i--;) { Offset o (curve_point (sols[i])); iv.unite (Interval (o[a], o[a])); @@ -187,9 +300,19 @@ Bezier::extent (Axis a) const return iv; } +Interval +Bezier::control_point_extent (Axis a) const +{ + Interval ext; + for (int i = CONTROL_COUNT; i--;) + ext.add_point (control_[i][a]); + + return ext; +} + /** Flip around axis A - */ +*/ void Bezier::scale (Real x, Real y) { @@ -220,7 +343,7 @@ Bezier::assert_sanity () const { for (int i = 0; i < CONTROL_COUNT; i++) assert (!isnan (control_[i].length ()) - && !isinf (control_[i].length ())); + && !isinf (control_[i].length ())); } void @@ -228,6 +351,54 @@ Bezier::reverse () { Bezier b2; for (int i = 0; i < CONTROL_COUNT; i++) - b2.control_[CONTROL_COUNT-i-1] = control_[i]; + b2.control_[CONTROL_COUNT - i - 1] = control_[i]; *this = b2; } + +/* + Subdivide a bezier at T into LEFT_PART and RIGHT_PART + using deCasteljau's algorithm. +*/ +void +Bezier::subdivide (Real t, Bezier *left_part, Bezier *right_part) const +{ + Offset p[CONTROL_COUNT][CONTROL_COUNT]; + + for (int i = 0; i < CONTROL_COUNT; i++) + p[i][CONTROL_COUNT - 1 ] = control_[i]; + for (int j = CONTROL_COUNT - 2; j >= 0; j--) + for (int i = 0; i < CONTROL_COUNT - 1; i++) + p[i][j] = p[i][j + 1] + t * (p[i + 1][j + 1] - p[i][j + 1]); + for (int i = 0; i < CONTROL_COUNT; i++) + { + left_part->control_[i] = p[0][CONTROL_COUNT - 1 - i]; + right_part->control_[i] = p[i][i]; + } +} + +/* + Extract a portion of a bezier from T_MIN to T_MAX +*/ + +Bezier +Bezier::extract (Real t_min, Real t_max) const +{ + if ((t_min < 0) || (t_max) > 1) + programming_error + ("bezier extract arguments outside of limits: curve may have bad shape"); + if (t_min >= t_max) + programming_error + ("lower bezier extract value not less than upper value: curve may have bad shape"); + Bezier bez1, bez2, bez3, bez4; + if (t_min == 0.0) + bez2 = *this; + else + subdivide (t_min, &bez1, &bez2); + if (t_max == 1.0) + return bez2; + else + { + bez2.subdivide ((t_max - t_min) / (1 - t_min), &bez3, &bez4); + return bez3; + } +}