2 This file is part of LilyPond, the GNU music typesetter.
4 Copyright (C) 1998--2011 Jan Nieuwenhuizen <janneke@gnu.org>
6 LilyPond is free software: you can redistribute it and/or modify
7 it under the terms of the GNU General Public License as published by
8 the Free Software Foundation, either version 3 of the License, or
9 (at your option) any later version.
11 LilyPond is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14 GNU General Public License for more details.
16 You should have received a copy of the GNU General Public License
17 along with LilyPond. If not, see <http://www.gnu.org/licenses/>.
22 #include "libc-extension.hh"
24 Real binomial_coefficient_3[] =
30 scale (vector<Offset> *array, Real x, Real y)
32 for (vsize i = 0; i < array->size (); i++)
34 (*array)[i][X_AXIS] = x * (*array)[i][X_AXIS];
35 (*array)[i][Y_AXIS] = y * (*array)[i][Y_AXIS];
40 rotate (vector<Offset> *array, Real phi)
42 Offset rot (complex_exp (Offset (0, phi)));
43 for (vsize i = 0; i < array->size (); i++)
44 (*array)[i] = complex_multiply (rot, (*array)[i]);
48 translate (vector<Offset> *array, Offset o)
50 for (vsize i = 0; i < array->size (); i++)
55 Formula of the bezier 3-spline
57 sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j) t^j
60 A is the axis of X coordinate.
64 Bezier::get_other_coordinate (Axis a, Real x) const
66 Axis other = Axis ((a + 1) % NO_AXES);
67 vector<Real> ts = solve_point (a, x);
71 programming_error ("no solution found for Bezier intersection");
76 Offset c = curve_point (ts[0]);
77 if (fabs (c[a] - x) > 1e-8)
78 programming_error ("bezier intersection not correct?");
81 return curve_coordinate (ts[0], other);
85 Bezier::curve_coordinate (Real t, Axis a) const
90 for (int i = 1; i < 4; i++)
91 one_min_tj[i] = one_min_tj[i - 1] * (1 - t);
94 for (int j = 0; j < 4; j++)
96 r += control_[j][a] * binomial_coefficient_3[j]
97 * tj * one_min_tj[3 - j];
106 Bezier::curve_point (Real t) const
111 for (int i = 1; i < 4; i++)
112 one_min_tj[i] = one_min_tj[i - 1] * (1 - t);
115 for (int j = 0; j < 4; j++)
117 o += control_[j] * binomial_coefficient_3[j]
118 * tj * one_min_tj[3 - j];
124 assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t)) < 1e-8);
125 assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t)) < 1e-8);
132 Cache binom (3, j) t^j (1-t)^{3-j}
134 struct Polynomial_cache
136 Polynomial terms_[4];
139 for (int j = 0; j <= 3; j++)
141 = binomial_coefficient_3[j]
142 * Polynomial::power (j, Polynomial (0, 1))
143 * Polynomial::power (3 - j, Polynomial (1, -1));
147 static Polynomial_cache poly_cache;
150 Bezier::polynomial (Axis a) const
154 for (int j = 0; j <= 3; j++)
156 q = poly_cache.terms_[j];
165 Remove all numbers outside [0, 1] from SOL
168 filter_solutions (vector<Real> sol)
170 for (vsize i = sol.size (); i--;)
171 if (sol[i] < 0 || sol[i] > 1)
172 sol.erase (sol.begin () + i);
177 find t such that derivative is proportional to DERIV
180 Bezier::solve_derivative (Offset deriv) const
182 Polynomial xp = polynomial (X_AXIS);
183 Polynomial yp = polynomial (Y_AXIS);
187 Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
189 return filter_solutions (combine.solve ());
193 Find t such that curve_point (t)[AX] == COORDINATE
196 Bezier::solve_point (Axis ax, Real coordinate) const
198 Polynomial p (polynomial (ax));
199 p.coefs_[0] -= coordinate;
201 vector<Real> sol (p.solve ());
202 return filter_solutions (sol);
206 Compute the bounding box dimensions in direction of A.
209 Bezier::extent (Axis a) const
211 int o = (a + 1) % NO_AXES;
215 vector<Real> sols (solve_derivative (d));
216 sols.push_back (1.0);
217 sols.push_back (0.0);
218 for (vsize i = sols.size (); i--;)
220 Offset o (curve_point (sols[i]));
221 iv.unite (Interval (o[a], o[a]));
227 Bezier::control_point_extent (Axis a) const
230 for (int i = CONTROL_COUNT; i--;)
231 ext.add_point (control_[i][a]);
240 Bezier::scale (Real x, Real y)
242 for (int i = CONTROL_COUNT; i--;)
244 control_[i][X_AXIS] = x * control_[i][X_AXIS];
245 control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
250 Bezier::rotate (Real phi)
252 Offset rot (complex_exp (Offset (0, phi)));
253 for (int i = 0; i < CONTROL_COUNT; i++)
254 control_[i] = complex_multiply (rot, control_[i]);
258 Bezier::translate (Offset o)
260 for (int i = 0; i < CONTROL_COUNT; i++)
265 Bezier::assert_sanity () const
267 for (int i = 0; i < CONTROL_COUNT; i++)
268 assert (!isnan (control_[i].length ())
269 && !isinf (control_[i].length ()));
276 for (int i = 0; i < CONTROL_COUNT; i++)
277 b2.control_[CONTROL_COUNT - i - 1] = control_[i];
282 Subdivide a bezier at T into LEFT_PART and RIGHT_PART
283 using deCasteljau's algorithm.
286 Bezier::subdivide (Real t, Bezier *left_part, Bezier *right_part) const
288 Offset p[CONTROL_COUNT][CONTROL_COUNT];
290 for (int i = 0; i < CONTROL_COUNT; i++)
291 p[i][CONTROL_COUNT - 1 ] = control_[i];
292 for (int j = CONTROL_COUNT - 2; j >= 0; j--)
293 for (int i = 0; i < CONTROL_COUNT - 1; i++)
294 p[i][j] = p[i][j + 1] + t * (p[i + 1][j + 1] - p[i][j + 1]);
295 for (int i = 0; i < CONTROL_COUNT; i++)
297 left_part->control_[i] = p[0][CONTROL_COUNT - 1 - i];
298 right_part->control_[i] = p[i][i];
303 Extract a portion of a bezier from T_MIN to T_MAX
307 Bezier::extract (Real t_min, Real t_max) const
309 if ((t_min < 0) || (t_max) > 1)
311 ("bezier extract arguments outside of limits: curve may have bad shape");
314 ("lower bezier extract value not less than upper value: curve may have bad shape");
315 Bezier bez1, bez2, bez3, bez4;
319 subdivide (t_min, &bez1, &bez2);
324 bez2.subdivide ((t_max - t_min) / (1 - t_min), &bez3, &bez4);