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.hh"
  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   A is the axis of X coordinate.
  65  */
  66
  67 Real
  68 Bezier::get_other_coordinate (Axis a,  Real x) const
  69 {
  70   Axis other = Axis ((a +1)%NO_AXES);
  71   Array<Real> ts = solve_point (a, x);
  72
  73   if (ts.size () == 0)
  74     {
  75       programming_error ("No solution found for Bezier intersection.");
  76       return 0.0;
  77     }
  78
  79   Offset c = curve_point (ts[0]);
  80
  81   if (fabs (c[a] - x) > 1e-8)
  82     programming_error ("Bezier intersection not correct?");
  83
  84   return c[other];
  85 }
  86
  87
  88 Offset
  89 Bezier::curve_point (Real t)const
  90 {
  91   Real tj = 1;
  92   Real one_min_tj = (1-t)* (1-t)* (1-t);
  93
  94   Offset o;
  95   for (int j=0 ; j < 4; j++)
  96     {
  97       o += control_[j] * binomial_coefficient (3, j)
  98         * pow (t,j) * pow (1-t, 3-j);
  99
 100       tj *= t;
 101       if (1-t)
 102         one_min_tj /= (1-t);
 103     }
 104
 105 #ifdef PARANOID
 106   assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t))< 1e-8);
 107   assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t))< 1e-8);
 108 #endif
 109
 110   return o;
 111 }
 112
 113
 114 Polynomial
 115 Bezier::polynomial (Axis a)const
 116 {
 117   Polynomial p (0.0);
 118   for (int j=0; j <= 3; j++)
 119     {
 120       p += (control_[j][a] *    binomial_coefficient (3, j))
 121         * Polynomial::power (j , Polynomial (0,1))*
 122         Polynomial::power (3 - j, Polynomial (1,-1));
 123     }
 124
 125   return p;
 126 }
 127
 128 /**
 129    Remove all numbers outside [0,1] from SOL
 130  */
 131 Array<Real>
 132 filter_solutions (Array<Real> sol)
 133 {
 134   for (int i = sol.size (); i--;)
 135     if (sol[i] < 0 || sol[i] >1)
 136       sol.del (i);
 137   return sol;
 138 }
 139
 140 /**
 141    find t such that derivative is proportional to DERIV
 142  */
 143 Array<Real>
 144 Bezier::solve_derivative (Offset deriv)const
 145 {
 146   Polynomial xp=polynomial (X_AXIS);
 147   Polynomial yp=polynomial (Y_AXIS);
 148   xp.differentiate ();
 149   yp.differentiate ();
 150
 151   Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
 152
 153   return filter_solutions (combine.solve ());
 154 }
 155
 156
 157 /*
 158   Find t such that curve_point (t)[AX] == COORDINATE
 159 */
 160 Array<Real>
 161 Bezier::solve_point (Axis ax, Real coordinate) const
 162 {
 163   Polynomial p (polynomial (ax));
 164   p.coefs_[0] -= coordinate;
 165
 166   Array<Real> sol (p.solve ());
 167   return filter_solutions (sol);
 168 }
 169
 170 /**
 171    Compute the bounding box dimensions in direction of A.
 172  */
 173 Interval
 174 Bezier::extent (Axis a)const
 175 {
 176   int o = (a+1)%NO_AXES;
 177   Offset d;
 178   d[Axis (o)] =1.0;
 179   Interval iv;
 180   Array<Real> sols (solve_derivative (d));
 181   sols.push (1.0);
 182   sols.push (0.0);
 183   for (int i= sols.size (); i--;)
 184     {
 185       Offset o (curve_point (sols[i]));
 186       iv.unite (Interval (o[a],o[a]));
 187     }
 188   return iv;
 189 }
 190
 191 /**
 192    Flip around axis A
 193  */
 194
 195 void
 196 Bezier::scale (Real x, Real y)
 197 {
 198   for (int i = CONTROL_COUNT; i--;)
 199     {
 200       control_[i][X_AXIS] = x * control_[i][X_AXIS];
 201       control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
 202     }
 203 }
 204
 205 void
 206 Bezier::rotate (Real phi)
 207 {
 208   Offset rot (complex_exp (Offset (0, phi)));
 209   for (int i = 0; i < CONTROL_COUNT; i++)
 210     control_[i] = complex_multiply (rot, control_[i]);
 211 }
 212
 213 void
 214 Bezier::translate (Offset o)
 215 {
 216   for (int i = 0; i < CONTROL_COUNT; i++)
 217     control_[i] += o;
 218 }
 219
 220 void
 221 Bezier::assert_sanity () const
 222 {
 223   for (int i=0; i < CONTROL_COUNT; i++)
 224     assert (!isnan (control_[i].length ())
 225             && !isinf (control_[i].length ()));
 226 }
 227
 228 void
 229 Bezier::reverse ()
 230 {
 231   Bezier b2;
 232   for (int i =0; i < CONTROL_COUNT; i++)
 233     b2.control_[CONTROL_COUNT-i-1] = control_[i];
 234   *this = b2;
 235 }