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--2002 Jan Nieuwenhuizen <janneke@gnu.org>
   7 */
   8
   9 #include <math.h>
  10
  11 #include "config.h"
  12 #include "warn.hh"
  13 #include "libc-extension.hh"
  14 #include "bezier.hh"
  15 #include "polynomial.hh"
  16
  17 Real
  18 binomial_coefficient (Real over , int under)
  19 {
  20   Real x = 1.0;
  21
  22   while (under)
  23     {
  24       x *= over / Real (under);
  25
  26       over  -= 1.0;
  27       under --;
  28     }
  29   return x;
  30 }
  31
  32 void
  33 scale (Array<Offset>* array, Real x , Real y)
  34 {
  35   for (int i = 0; i < array->size (); i++)
  36     {
  37       (*array)[i][X_AXIS] = x* (*array)[i][X_AXIS];
  38       (*array)[i][Y_AXIS] = y* (*array)[i][Y_AXIS];
  39     }
  40 }
  41
  42 void
  43 rotate (Array<Offset>* array, Real phi)
  44 {
  45   Offset rot (complex_exp (Offset (0, phi)));
  46   for (int i = 0; i < array->size (); i++)
  47     (*array)[i] = complex_multiply (rot, (*array)[i]);
  48 }
  49
  50 void
  51 translate (Array<Offset>* array, Offset o)
  52 {
  53   for (int i = 0; i < array->size (); i++)
  54     (*array)[i] += o;
  55 }
  56
  57 /*
  58
  59   Formula of the bezier 3-spline
  60
  61   sum_{j=0}^3 (3 over j) z_j (1-t)^ (3-j)  t^j
  62  */
  63
  64 Real
  65 Bezier::get_other_coordinate (Axis a,  Real x) const
  66 {
  67   Axis other = Axis ((a +1)%NO_AXES);
  68   Array<Real> ts = solve_point (a, x);
  69
  70   if (ts.size () == 0)
  71     {
  72       programming_error ("No solution found for Bezier intersection.");
  73       return 0.0;
  74     }
  75
  76   Offset c = curve_point (ts[0]);
  77   assert (fabs (c[a] - x) < 1e-8);
  78
  79   return c[other];
  80 }
  81
  82
  83 Offset
  84 Bezier::curve_point (Real t)const
  85 {
  86   Real tj = 1;
  87   Real one_min_tj = (1-t)* (1-t)* (1-t);
  88
  89   Offset o;
  90   for (int j=0 ; j < 4; j++)
  91     {
  92       o += control_[j] * binomial_coefficient (3, j)
  93         * pow (t,j) * pow (1-t, 3-j);
  94
  95       tj *= t;
  96       if (1-t)
  97         one_min_tj /= (1-t);
  98     }
  99
 100 #ifdef PARANOID
 101   assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t))< 1e-8);
 102   assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t))< 1e-8);
 103 #endif
 104
 105   return o;
 106 }
 107
 108
 109 Polynomial
 110 Bezier::polynomial (Axis a)const
 111 {
 112   Polynomial p (0.0);
 113   for (int j=0; j <= 3; j++)
 114     {
 115       p += (control_[j][a] *    binomial_coefficient (3, j))
 116         * Polynomial::power (j , Polynomial (0,1))*
 117         Polynomial::power (3 - j, Polynomial (1,-1));
 118     }
 119
 120   return p;
 121 }
 122
 123 /**
 124    Remove all numbers outside [0,1] from SOL
 125  */
 126 Array<Real>
 127 filter_solutions (Array<Real> sol)
 128 {
 129   for (int i = sol.size (); i--;)
 130     if (sol[i] < 0 || sol[i] >1)
 131       sol.del (i);
 132   return sol;
 133 }
 134
 135 /**
 136    find t such that derivative is proportional to DERIV
 137  */
 138 Array<Real>
 139 Bezier::solve_derivative (Offset deriv)const
 140 {
 141   Polynomial xp=polynomial (X_AXIS);
 142   Polynomial yp=polynomial (Y_AXIS);
 143   xp.differentiate ();
 144   yp.differentiate ();
 145
 146   Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
 147
 148   return filter_solutions (combine.solve ());
 149 }
 150
 151
 152 /*
 153   Find t such that curve_point (t)[AX] == COORDINATE
 154 */
 155 Array<Real>
 156 Bezier::solve_point (Axis ax, Real coordinate) const
 157 {
 158   Polynomial p (polynomial (ax));
 159   p.coefs_[0] -= coordinate;
 160
 161   Array<Real> sol (p.solve ());
 162   return filter_solutions (sol);
 163 }
 164
 165 /**
 166    Compute the bounding box dimensions in direction of A.
 167  */
 168 Interval
 169 Bezier::extent (Axis a)const
 170 {
 171   int o = (a+1)%NO_AXES;
 172   Offset d;
 173   d[Axis (o)] =1.0;
 174   Interval iv;
 175   Array<Real> sols (solve_derivative (d));
 176   sols.push (1.0);
 177   sols.push (0.0);
 178   for (int i= sols.size (); i--;)
 179     {
 180       Offset o (curve_point (sols[i]));
 181       iv.unite (Interval (o[a],o[a]));
 182     }
 183   return iv;
 184 }
 185
 186 /**
 187    Flip around axis A
 188  */
 189
 190 void
 191 Bezier::scale (Real x, Real y)
 192 {
 193   for (int i = CONTROL_COUNT; i--;)
 194     {
 195       control_[i][X_AXIS] = x * control_[i][X_AXIS];
 196       control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
 197     }
 198 }
 199
 200 void
 201 Bezier::rotate (Real phi)
 202 {
 203   Offset rot (complex_exp (Offset (0, phi)));
 204   for (int i = 0; i < CONTROL_COUNT; i++)
 205     control_[i] = complex_multiply (rot, control_[i]);
 206 }
 207
 208 void
 209 Bezier::translate (Offset o)
 210 {
 211   for (int i = 0; i < CONTROL_COUNT; i++)
 212     control_[i] += o;
 213 }
 214
 215 void
 216 Bezier::assert_sanity () const
 217 {
 218   for (int i=0; i < CONTROL_COUNT; i++)
 219     assert (!isnan (control_[i].length ())
 220             && !isinf (control_[i].length ()));
 221 }
 222
 223 void
 224 Bezier::reverse ()
 225 {
 226   Bezier b2;
 227   for (int i =0; i < CONTROL_COUNT; i++)
 228     b2.control_[CONTROL_COUNT-i-1] = control_[i];
 229   *this = b2;
 230 }