lily/bezier.cc

   1 /*
   2   bezier.cc -- implement Bezier and Bezier_bow
   3
   4   source file of the GNU LilyPond music typesetter
   5
   6   (c) 1998--2009 Jan Nieuwenhuizen <janneke@gnu.org>
   7 */
   8
   9 #include "bezier.hh"
  10 #include "warn.hh"
  11 #include "libc-extension.hh"
  12
  13 Real binomial_coefficient_3[] = {
  14   1, 3, 3, 1
  15 };
  16
  17 void
  18 scale (vector<Offset> *array, Real x, Real y)
  19 {
  20   for (vsize i = 0; i < array->size (); i++)
  21     {
  22       (*array)[i][X_AXIS] = x * (*array)[i][X_AXIS];
  23       (*array)[i][Y_AXIS] = y * (*array)[i][Y_AXIS];
  24     }
  25 }
  26
  27 void
  28 rotate (vector<Offset> *array, Real phi)
  29 {
  30   Offset rot (complex_exp (Offset (0, phi)));
  31   for (vsize i = 0; i < array->size (); i++)
  32     (*array)[i] = complex_multiply (rot, (*array)[i]);
  33 }
  34
  35 void
  36 translate (vector<Offset> *array, Offset o)
  37 {
  38   for (vsize i = 0; i < array->size (); i++)
  39     (*array)[i] += o;
  40 }
  41
  42 /*
  43   Formula of the bezier 3-spline
  44
  45   sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j)  t^j
  46
  47
  48   A is the axis of X coordinate.
  49 */
  50
  51 Real
  52 Bezier::get_other_coordinate (Axis a, Real x) const
  53 {
  54   Axis other = Axis ((a +1) % NO_AXES);
  55   vector<Real> ts = solve_point (a, x);
  56
  57   if (ts.size () == 0)
  58     {
  59       programming_error ("no solution found for Bezier intersection");
  60       return 0.0;
  61     }
  62
  63 #ifdef PARANOID
  64   Offset c = curve_point (ts[0]);
  65   if (fabs (c[a] - x) > 1e-8)
  66     programming_error ("bezier intersection not correct?");
  67 #endif
  68
  69   return curve_coordinate (ts[0], other);
  70 }
  71
  72 Real
  73 Bezier::curve_coordinate (Real t, Axis a) const
  74 {
  75   Real tj = 1;
  76   Real one_min_tj[4];
  77   one_min_tj[0] = 1;
  78   for (int i = 1; i < 4; i++)
  79     one_min_tj[i] = one_min_tj[i - 1] * (1 - t);
  80
  81   Real r = 0.0;
  82   for (int j = 0; j < 4; j++)
  83     {
  84       r += control_[j][a] * binomial_coefficient_3[j]
  85         * tj * one_min_tj[3 - j];
  86
  87       tj *= t;
  88     }
  89
  90   return r;
  91 }
  92
  93 Offset
  94 Bezier::curve_point (Real t) const
  95 {
  96   Real tj = 1;
  97   Real one_min_tj[4];
  98   one_min_tj[0] = 1;
  99   for (int i = 1; i < 4; i++)
 100     one_min_tj[i] = one_min_tj[i - 1] * (1 - t);
 101
 102   Offset o;
 103   for (int j = 0; j < 4; j++)
 104     {
 105       o += control_[j] * binomial_coefficient_3[j]
 106         * tj * one_min_tj[3 - j];
 107
 108       tj *= t;
 109     }
 110
 111 #ifdef PARANOID
 112   assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t)) < 1e-8);
 113   assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t)) < 1e-8);
 114 #endif
 115
 116   return o;
 117 }
 118
 119 /*
 120   Cache binom (3, j) t^j (1-t)^{3-j}
 121 */
 122 struct Polynomial_cache {
 123   Polynomial terms_[4];
 124   Polynomial_cache ()
 125   {
 126     for (int j = 0; j <= 3; j++)
 127       terms_[j]
 128         = binomial_coefficient_3[j]
 129         * Polynomial::power (j, Polynomial (0, 1))
 130         * Polynomial::power (3 - j, Polynomial (1, -1));
 131   }
 132 };
 133
 134 static Polynomial_cache poly_cache;
 135
 136 Polynomial
 137 Bezier::polynomial (Axis a) const
 138 {
 139   Polynomial p (0.0);
 140   Polynomial q;
 141   for (int j = 0; j <= 3; j++)
 142     {
 143       q = poly_cache.terms_[j];
 144       q *= control_[j][a];
 145       p += q;
 146     }
 147
 148   return p;
 149 }
 150
 151 /**
 152    Remove all numbers outside [0, 1] from SOL
 153 */
 154 vector<Real>
 155 filter_solutions (vector<Real> sol)
 156 {
 157   for (vsize i = sol.size (); i--;)
 158     if (sol[i] < 0 || sol[i] > 1)
 159       sol.erase (sol.begin () + i);
 160   return sol;
 161 }
 162
 163 /**
 164    find t such that derivative is proportional to DERIV
 165 */
 166 vector<Real>
 167 Bezier::solve_derivative (Offset deriv) const
 168 {
 169   Polynomial xp = polynomial (X_AXIS);
 170   Polynomial yp = polynomial (Y_AXIS);
 171   xp.differentiate ();
 172   yp.differentiate ();
 173
 174   Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
 175
 176   return filter_solutions (combine.solve ());
 177 }
 178
 179 /*
 180   Find t such that curve_point (t)[AX] == COORDINATE
 181 */
 182 vector<Real>
 183 Bezier::solve_point (Axis ax, Real coordinate) const
 184 {
 185   Polynomial p (polynomial (ax));
 186   p.coefs_[0] -= coordinate;
 187
 188   vector<Real> sol (p.solve ());
 189   return filter_solutions (sol);
 190 }
 191
 192 /**
 193    Compute the bounding box dimensions in direction of A.
 194 */
 195 Interval
 196 Bezier::extent (Axis a) const
 197 {
 198   int o = (a + 1)%NO_AXES;
 199   Offset d;
 200   d[Axis (o)] = 1.0;
 201   Interval iv;
 202   vector<Real> sols (solve_derivative (d));
 203   sols.push_back (1.0);
 204   sols.push_back (0.0);
 205   for (vsize i = sols.size (); i--;)
 206     {
 207       Offset o (curve_point (sols[i]));
 208       iv.unite (Interval (o[a], o[a]));
 209     }
 210   return iv;
 211 }
 212
 213 Interval
 214 Bezier::control_point_extent (Axis a) const
 215 {
 216   Interval ext;
 217   for (int i = CONTROL_COUNT; i--;)
 218     ext.add_point (control_[i][a]);
 219
 220   return ext;
 221 }
 222
 223
 224 /**
 225    Flip around axis A
 226 */
 227 void
 228 Bezier::scale (Real x, Real y)
 229 {
 230   for (int i = CONTROL_COUNT; i--;)
 231     {
 232       control_[i][X_AXIS] = x * control_[i][X_AXIS];
 233       control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
 234     }
 235 }
 236
 237 void
 238 Bezier::rotate (Real phi)
 239 {
 240   Offset rot (complex_exp (Offset (0, phi)));
 241   for (int i = 0; i < CONTROL_COUNT; i++)
 242     control_[i] = complex_multiply (rot, control_[i]);
 243 }
 244
 245 void
 246 Bezier::translate (Offset o)
 247 {
 248   for (int i = 0; i < CONTROL_COUNT; i++)
 249     control_[i] += o;
 250 }
 251
 252 void
 253 Bezier::assert_sanity () const
 254 {
 255   for (int i = 0; i < CONTROL_COUNT; i++)
 256     assert (!isnan (control_[i].length ())
 257             && !isinf (control_[i].length ()));
 258 }
 259
 260 void
 261 Bezier::reverse ()
 262 {
 263   Bezier b2;
 264   for (int i = 0; i < CONTROL_COUNT; i++)
 265     b2.control_[CONTROL_COUNT - i - 1] = control_[i];
 266   *this = b2;
 267 }
 268
 269
 270 /*
 271   Subdivide a bezier at T into LEFT_PART and RIGHT_PART
 272   using deCasteljau's algorithm.
 273 */
 274 void
 275 Bezier::subdivide (Real t, Bezier *left_part, Bezier *right_part) const
 276 {
 277   Offset p[CONTROL_COUNT][CONTROL_COUNT];
 278
 279   for (int i = 0; i < CONTROL_COUNT ; i++)
 280     p[i][CONTROL_COUNT - 1 ] = control_[i];
 281   for (int j = CONTROL_COUNT - 2; j >= 0 ; j--)
 282   for (int i = 0; i < CONTROL_COUNT -1; i++)
 283     p[i][j] = p[i][j+1] + t * (p[i+1][j+1] - p[i][j+1]);
 284   for (int i = 0; i < CONTROL_COUNT; i++)
 285     {
 286       left_part->control_[i]=p[0][CONTROL_COUNT - 1 - i];
 287       right_part->control_[i]=p[i][i];
 288     }
 289 }
 290
 291 /*
 292   Extract a portion of a bezier from T_MIN to T_MAX
 293 */
 294
 295 Bezier
 296 Bezier::extract (Real t_min, Real t_max) const
 297 {
 298   if ((t_min < 0) || (t_max) > 1)
 299     programming_error
 300       ("bezier extract arguments outside of limits: curve may have bad shape");
 301   if (t_min >= t_max)
 302     programming_error
 303       ("lower bezier extract value not less than upper value: curve may have bad shape");
 304   Bezier bez1, bez2, bez3, bez4;
 305   if (t_min == 0.0)
 306     bez2 = *this;
 307   else
 308       subdivide (t_min, &bez1, &bez2);
 309   if (t_max == 1.0)
 310       return bez2;
 311   else
 312    {
 313      bez2.subdivide ((t_max-t_min)/(1-t_min), &bez3, &bez4);
 314      return bez3;
 315   }
 316 }