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--2004 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
  78   if (fabs (c[a] - x) > 1e-8)
  79     programming_error ("Bezier intersection not correct?");
  80
  81   return c[other];
  82 }
  83
  84
  85 Offset
  86 Bezier::curve_point (Real t)const
  87 {
  88   Real tj = 1;
  89   Real one_min_tj = (1-t)* (1-t)* (1-t);
  90
  91   Offset o;
  92   for (int j=0 ; j < 4; j++)
  93     {
  94       o += control_[j] * binomial_coefficient (3, j)
  95         * pow (t,j) * pow (1-t, 3-j);
  96
  97       tj *= t;
  98       if (1-t)
  99         one_min_tj /= (1-t);
 100     }
 101
 102 #ifdef PARANOID
 103   assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t))< 1e-8);
 104   assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t))< 1e-8);
 105 #endif
 106
 107   return o;
 108 }
 109
 110
 111 Polynomial
 112 Bezier::polynomial (Axis a)const
 113 {
 114   Polynomial p (0.0);
 115   for (int j=0; j <= 3; j++)
 116     {
 117       p += (control_[j][a] *    binomial_coefficient (3, j))
 118         * Polynomial::power (j , Polynomial (0,1))*
 119         Polynomial::power (3 - j, Polynomial (1,-1));
 120     }
 121
 122   return p;
 123 }
 124
 125 /**
 126    Remove all numbers outside [0,1] from SOL
 127  */
 128 Array<Real>
 129 filter_solutions (Array<Real> sol)
 130 {
 131   for (int i = sol.size (); i--;)
 132     if (sol[i] < 0 || sol[i] >1)
 133       sol.del (i);
 134   return sol;
 135 }
 136
 137 /**
 138    find t such that derivative is proportional to DERIV
 139  */
 140 Array<Real>
 141 Bezier::solve_derivative (Offset deriv)const
 142 {
 143   Polynomial xp=polynomial (X_AXIS);
 144   Polynomial yp=polynomial (Y_AXIS);
 145   xp.differentiate ();
 146   yp.differentiate ();
 147
 148   Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
 149
 150   return filter_solutions (combine.solve ());
 151 }
 152
 153
 154 /*
 155   Find t such that curve_point (t)[AX] == COORDINATE
 156 */
 157 Array<Real>
 158 Bezier::solve_point (Axis ax, Real coordinate) const
 159 {
 160   Polynomial p (polynomial (ax));
 161   p.coefs_[0] -= coordinate;
 162
 163   Array<Real> sol (p.solve ());
 164   return filter_solutions (sol);
 165 }
 166
 167 /**
 168    Compute the bounding box dimensions in direction of A.
 169  */
 170 Interval
 171 Bezier::extent (Axis a)const
 172 {
 173   int o = (a+1)%NO_AXES;
 174   Offset d;
 175   d[Axis (o)] =1.0;
 176   Interval iv;
 177   Array<Real> sols (solve_derivative (d));
 178   sols.push (1.0);
 179   sols.push (0.0);
 180   for (int i= sols.size (); i--;)
 181     {
 182       Offset o (curve_point (sols[i]));
 183       iv.unite (Interval (o[a],o[a]));
 184     }
 185   return iv;
 186 }
 187
 188 /**
 189    Flip around axis A
 190  */
 191
 192 void
 193 Bezier::scale (Real x, Real y)
 194 {
 195   for (int i = CONTROL_COUNT; i--;)
 196     {
 197       control_[i][X_AXIS] = x * control_[i][X_AXIS];
 198       control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
 199     }
 200 }
 201
 202 void
 203 Bezier::rotate (Real phi)
 204 {
 205   Offset rot (complex_exp (Offset (0, phi)));
 206   for (int i = 0; i < CONTROL_COUNT; i++)
 207     control_[i] = complex_multiply (rot, control_[i]);
 208 }
 209
 210 void
 211 Bezier::translate (Offset o)
 212 {
 213   for (int i = 0; i < CONTROL_COUNT; i++)
 214     control_[i] += o;
 215 }
 216
 217 void
 218 Bezier::assert_sanity () const
 219 {
 220   for (int i=0; i < CONTROL_COUNT; i++)
 221     assert (!isnan (control_[i].length ())
 222             && !isinf (control_[i].length ()));
 223 }
 224
 225 void
 226 Bezier::reverse ()
 227 {
 228   Bezier b2;
 229   for (int i =0; i < CONTROL_COUNT; i++)
 230     b2.control_[CONTROL_COUNT-i-1] = control_[i];
 231   *this = b2;
 232 }