lily/bezier.cc

   1 /*
   2   This file is part of LilyPond, the GNU music typesetter.
   3
   4   Copyright (C) 1998--2010 Jan Nieuwenhuizen <janneke@gnu.org>
   5
   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.
  10
  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.
  15
  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/>.
  18 */
  19
  20 #include "bezier.hh"
  21 #include "warn.hh"
  22 #include "libc-extension.hh"
  23
  24 Real binomial_coefficient_3[] = {
  25   1, 3, 3, 1
  26 };
  27
  28 void
  29 scale (vector<Offset> *array, Real x, Real y)
  30 {
  31   for (vsize i = 0; i < array->size (); i++)
  32     {
  33       (*array)[i][X_AXIS] = x * (*array)[i][X_AXIS];
  34       (*array)[i][Y_AXIS] = y * (*array)[i][Y_AXIS];
  35     }
  36 }
  37
  38 void
  39 rotate (vector<Offset> *array, Real phi)
  40 {
  41   Offset rot (complex_exp (Offset (0, phi)));
  42   for (vsize i = 0; i < array->size (); i++)
  43     (*array)[i] = complex_multiply (rot, (*array)[i]);
  44 }
  45
  46 void
  47 translate (vector<Offset> *array, Offset o)
  48 {
  49   for (vsize i = 0; i < array->size (); i++)
  50     (*array)[i] += o;
  51 }
  52
  53 /*
  54   Formula of the bezier 3-spline
  55
  56   sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j)  t^j
  57
  58
  59   A is the axis of X coordinate.
  60 */
  61
  62 Real
  63 Bezier::get_other_coordinate (Axis a, Real x) const
  64 {
  65   Axis other = Axis ((a +1) % NO_AXES);
  66   vector<Real> ts = solve_point (a, x);
  67
  68   if (ts.size () == 0)
  69     {
  70       programming_error ("no solution found for Bezier intersection");
  71       return 0.0;
  72     }
  73
  74 #ifdef PARANOID
  75   Offset c = curve_point (ts[0]);
  76   if (fabs (c[a] - x) > 1e-8)
  77     programming_error ("bezier intersection not correct?");
  78 #endif
  79
  80   return curve_coordinate (ts[0], other);
  81 }
  82
  83 Real
  84 Bezier::curve_coordinate (Real t, Axis a) const
  85 {
  86   Real tj = 1;
  87   Real one_min_tj[4];
  88   one_min_tj[0] = 1;
  89   for (int i = 1; i < 4; i++)
  90     one_min_tj[i] = one_min_tj[i - 1] * (1 - t);
  91
  92   Real r = 0.0;
  93   for (int j = 0; j < 4; j++)
  94     {
  95       r += control_[j][a] * binomial_coefficient_3[j]
  96         * tj * one_min_tj[3 - j];
  97
  98       tj *= t;
  99     }
 100
 101   return r;
 102 }
 103
 104 Offset
 105 Bezier::curve_point (Real t) const
 106 {
 107   Real tj = 1;
 108   Real one_min_tj[4];
 109   one_min_tj[0] = 1;
 110   for (int i = 1; i < 4; i++)
 111     one_min_tj[i] = one_min_tj[i - 1] * (1 - t);
 112
 113   Offset o;
 114   for (int j = 0; j < 4; j++)
 115     {
 116       o += control_[j] * binomial_coefficient_3[j]
 117         * tj * one_min_tj[3 - j];
 118
 119       tj *= t;
 120     }
 121
 122 #ifdef PARANOID
 123   assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t)) < 1e-8);
 124   assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t)) < 1e-8);
 125 #endif
 126
 127   return o;
 128 }
 129
 130 /*
 131   Cache binom (3, j) t^j (1-t)^{3-j}
 132 */
 133 struct Polynomial_cache {
 134   Polynomial terms_[4];
 135   Polynomial_cache ()
 136   {
 137     for (int j = 0; j <= 3; j++)
 138       terms_[j]
 139         = binomial_coefficient_3[j]
 140         * Polynomial::power (j, Polynomial (0, 1))
 141         * Polynomial::power (3 - j, Polynomial (1, -1));
 142   }
 143 };
 144
 145 static Polynomial_cache poly_cache;
 146
 147 Polynomial
 148 Bezier::polynomial (Axis a) const
 149 {
 150   Polynomial p (0.0);
 151   Polynomial q;
 152   for (int j = 0; j <= 3; j++)
 153     {
 154       q = poly_cache.terms_[j];
 155       q *= control_[j][a];
 156       p += q;
 157     }
 158
 159   return p;
 160 }
 161
 162 /**
 163    Remove all numbers outside [0, 1] from SOL
 164 */
 165 vector<Real>
 166 filter_solutions (vector<Real> sol)
 167 {
 168   for (vsize i = sol.size (); i--;)
 169     if (sol[i] < 0 || sol[i] > 1)
 170       sol.erase (sol.begin () + i);
 171   return sol;
 172 }
 173
 174 /**
 175    find t such that derivative is proportional to DERIV
 176 */
 177 vector<Real>
 178 Bezier::solve_derivative (Offset deriv) const
 179 {
 180   Polynomial xp = polynomial (X_AXIS);
 181   Polynomial yp = polynomial (Y_AXIS);
 182   xp.differentiate ();
 183   yp.differentiate ();
 184
 185   Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
 186
 187   return filter_solutions (combine.solve ());
 188 }
 189
 190 /*
 191   Find t such that curve_point (t)[AX] == COORDINATE
 192 */
 193 vector<Real>
 194 Bezier::solve_point (Axis ax, Real coordinate) const
 195 {
 196   Polynomial p (polynomial (ax));
 197   p.coefs_[0] -= coordinate;
 198
 199   vector<Real> sol (p.solve ());
 200   return filter_solutions (sol);
 201 }
 202
 203 /**
 204    Compute the bounding box dimensions in direction of A.
 205 */
 206 Interval
 207 Bezier::extent (Axis a) const
 208 {
 209   int o = (a + 1)%NO_AXES;
 210   Offset d;
 211   d[Axis (o)] = 1.0;
 212   Interval iv;
 213   vector<Real> sols (solve_derivative (d));
 214   sols.push_back (1.0);
 215   sols.push_back (0.0);
 216   for (vsize i = sols.size (); i--;)
 217     {
 218       Offset o (curve_point (sols[i]));
 219       iv.unite (Interval (o[a], o[a]));
 220     }
 221   return iv;
 222 }
 223
 224 Interval
 225 Bezier::control_point_extent (Axis a) const
 226 {
 227   Interval ext;
 228   for (int i = CONTROL_COUNT; i--;)
 229     ext.add_point (control_[i][a]);
 230
 231   return ext;
 232 }
 233
 234
 235 /**
 236    Flip around axis A
 237 */
 238 void
 239 Bezier::scale (Real x, Real y)
 240 {
 241   for (int i = CONTROL_COUNT; i--;)
 242     {
 243       control_[i][X_AXIS] = x * control_[i][X_AXIS];
 244       control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
 245     }
 246 }
 247
 248 void
 249 Bezier::rotate (Real phi)
 250 {
 251   Offset rot (complex_exp (Offset (0, phi)));
 252   for (int i = 0; i < CONTROL_COUNT; i++)
 253     control_[i] = complex_multiply (rot, control_[i]);
 254 }
 255
 256 void
 257 Bezier::translate (Offset o)
 258 {
 259   for (int i = 0; i < CONTROL_COUNT; i++)
 260     control_[i] += o;
 261 }
 262
 263 void
 264 Bezier::assert_sanity () const
 265 {
 266   for (int i = 0; i < CONTROL_COUNT; i++)
 267     assert (!isnan (control_[i].length ())
 268             && !isinf (control_[i].length ()));
 269 }
 270
 271 void
 272 Bezier::reverse ()
 273 {
 274   Bezier b2;
 275   for (int i = 0; i < CONTROL_COUNT; i++)
 276     b2.control_[CONTROL_COUNT - i - 1] = control_[i];
 277   *this = b2;
 278 }
 279
 280
 281 /*
 282   Subdivide a bezier at T into LEFT_PART and RIGHT_PART
 283   using deCasteljau's algorithm.
 284 */
 285 void
 286 Bezier::subdivide (Real t, Bezier *left_part, Bezier *right_part) const
 287 {
 288   Offset p[CONTROL_COUNT][CONTROL_COUNT];
 289
 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++)
 296     {
 297       left_part->control_[i]=p[0][CONTROL_COUNT - 1 - i];
 298       right_part->control_[i]=p[i][i];
 299     }
 300 }
 301
 302 /*
 303   Extract a portion of a bezier from T_MIN to T_MAX
 304 */
 305
 306 Bezier
 307 Bezier::extract (Real t_min, Real t_max) const
 308 {
 309   if ((t_min < 0) || (t_max) > 1)
 310     programming_error
 311       ("bezier extract arguments outside of limits: curve may have bad shape");
 312   if (t_min >= t_max)
 313     programming_error
 314       ("lower bezier extract value not less than upper value: curve may have bad shape");
 315   Bezier bez1, bez2, bez3, bez4;
 316   if (t_min == 0.0)
 317     bez2 = *this;
 318   else
 319       subdivide (t_min, &bez1, &bez2);
 320   if (t_max == 1.0)
 321       return bez2;
 322   else
 323    {
 324      bez2.subdivide ((t_max-t_min)/(1-t_min), &bez3, &bez4);
 325      return bez3;
 326   }
 327 }