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--2001 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>* arr_p, Real x , Real y)
  34 {
  35   for (int i = 0; i < arr_p->size (); i++)
  36     {
  37       (*arr_p)[i][X_AXIS] = x* (*arr_p)[i][X_AXIS];
  38       (*arr_p)[i][Y_AXIS] = y* (*arr_p)[i][Y_AXIS];
  39     }
  40 }
  41
  42 void
  43 rotate (Array<Offset>* arr_p, Real phi)
  44 {
  45   Offset rot (complex_exp (Offset (0, phi)));
  46   for (int i = 0; i < arr_p->size (); i++)
  47     (*arr_p)[i] = complex_multiply (rot, (*arr_p)[i]);
  48 }
  49
  50 void
  51 translate (Array<Offset>* arr_p, Offset o)
  52 {
  53   for (int i = 0; i < arr_p->size (); i++)
  54     (*arr_p)[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]
 116         * Polynomial::power (j , Polynomial (0,1))*
 117         Polynomial::power (3 - j, Polynomial (1,-1))*
 118         binomial_coefficient (3, j);
 119     }
 120
 121   return p;
 122 }
 123
 124 /**
 125    Remove all numbers outside [0,1] from SOL
 126  */
 127 Array<Real>
 128 filter_solutions (Array<Real> sol)
 129 {
 130   for (int i = sol.size (); i--;)
 131     if (sol[i] < 0 || sol[i] >1)
 132       sol.del (i);
 133   return sol;
 134 }
 135
 136 /**
 137    find t such that derivative is proportional to DERIV
 138  */
 139 Array<Real>
 140 Bezier::solve_derivative (Offset deriv)const
 141 {
 142   Polynomial xp=polynomial (X_AXIS);
 143   Polynomial yp=polynomial (Y_AXIS);
 144   xp.differentiate ();
 145   yp.differentiate ();
 146
 147   Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
 148
 149   return filter_solutions (combine.solve ());
 150 }
 151
 152
 153 /*
 154   Find t such that curve_point (t)[AX] == COORDINATE
 155 */
 156 Array<Real>
 157 Bezier::solve_point (Axis ax, Real coordinate) const
 158 {
 159   Polynomial p (polynomial (ax));
 160   p.coefs_[0] -= coordinate;
 161
 162   Array<Real> sol (p.solve ());
 163   return filter_solutions (sol);
 164 }
 165
 166 /**
 167    Compute the bounding box dimensions in direction of A.
 168  */
 169 Interval
 170 Bezier::extent (Axis a)const
 171 {
 172   int o = (a+1)%NO_AXES;
 173   Offset d;
 174   d[Axis (o)] =1.0;
 175   Interval iv;
 176   Array<Real> sols (solve_derivative (d));
 177   sols.push (1.0);
 178   sols.push (0.0);
 179   for (int i= sols.size (); i--;)
 180     {
 181       Offset o (curve_point (sols[i]));
 182       iv.unite (Interval (o[a],o[a]));
 183     }
 184   return iv;
 185 }
 186
 187 /**
 188    Flip around axis A
 189  */
 190
 191 void
 192 Bezier::scale (Real x, Real y)
 193 {
 194   for (int i = CONTROL_COUNT; i--;)
 195     {
 196       control_[i][X_AXIS] = x * control_[i][X_AXIS];
 197       control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
 198     }
 199 }
 200
 201 void
 202 Bezier::rotate (Real phi)
 203 {
 204   Offset rot (complex_exp (Offset (0, phi)));
 205   for (int i = 0; i < CONTROL_COUNT; i++)
 206     control_[i] = complex_multiply (rot, control_[i]);
 207 }
 208
 209 void
 210 Bezier::translate (Offset o)
 211 {
 212   for (int i = 0; i < CONTROL_COUNT; i++)
 213     control_[i] += o;
 214 }
 215
 216 void
 217 Bezier::assert_sanity () const
 218 {
 219   for (int i=0; i < CONTROL_COUNT; i++)
 220     assert (!isnan (control_[i].length ())
 221             && !isinf (control_[i].length ()));
 222 }
 223
 224 void
 225 Bezier::reverse ()
 226 {
 227   Bezier b2;
 228   for (int i =0; i < CONTROL_COUNT; i++)
 229     b2.control_[CONTROL_COUNT-i-1] = control_[i];
 230   *this = b2;
 231 }