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--2000 Jan Nieuwenhuizen <janneke@gnu.org>
   7 */
   8
   9 #include <math.h>
  10 #include "config.h"
  11
  12 #include "libc-extension.hh"
  13 #include "bezier.hh"
  14 #include "polynomial.hh"
  15
  16 Real
  17 binomial_coefficient (Real over , int under)
  18 {
  19   Real x = 1.0;
  20
  21   while (under)
  22     {
  23       x *= over / Real (under);
  24
  25       over  -= 1.0;
  26       under --;
  27     }
  28   return x;
  29 }
  30
  31 void
  32 scale (Array<Offset>* arr_p, Real x , Real y)
  33 {
  34   for (int i = 0; i < arr_p->size (); i++)
  35     {
  36       (*arr_p)[i][X_AXIS] = x* (*arr_p)[i][X_AXIS];
  37       (*arr_p)[i][Y_AXIS] = y* (*arr_p)[i][Y_AXIS];
  38     }
  39 }
  40
  41 void
  42 rotate (Array<Offset>* arr_p, Real phi)
  43 {
  44   Offset rot (complex_exp (Offset (0, phi)));
  45   for (int i = 0; i < arr_p->size (); i++)
  46     (*arr_p)[i] = complex_multiply (rot, (*arr_p)[i]);
  47 }
  48
  49 void
  50 translate (Array<Offset>* arr_p, Offset o)
  51 {
  52   for (int i = 0; i < arr_p->size (); i++)
  53     (*arr_p)[i] += o;
  54 }
  55
  56 /*
  57
  58   Formula of the bezier 3-spline
  59
  60   sum_{j=0}^3  (3 over j) z_j (1-t)^(3-j)  t^j
  61  */
  62
  63 Real
  64 Bezier::get_other_coordinate (Axis a,  Real x) const
  65 {
  66   Axis other = Axis ((a +1)%NO_AXES);
  67   Array<Real> ts = solve_point (a, x);
  68
  69   Offset c = curve_point (ts[0]);
  70   assert (fabs (c[a] - x) < 1e-8);
  71
  72   return c[other];
  73 }
  74
  75
  76 Offset
  77 Bezier::curve_point (Real t)const
  78 {
  79   Real tj = 1;
  80   Real one_min_tj =  (1-t)*(1-t)*(1-t);
  81
  82   Offset o;
  83   for (int j=0 ; j < 4; j++)
  84     {
  85       o += control_[j] * binomial_coefficient (3, j)
  86         * pow (t,j) * pow (1-t, 3-j);
  87
  88       tj *= t;
  89       if (1-t)
  90         one_min_tj /= (1-t);
  91     }
  92
  93 #ifdef PARANOID
  94   assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t))< 1e-8);
  95   assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t))< 1e-8);
  96 #endif
  97
  98   return o;
  99 }
 100
 101
 102 Polynomial
 103 Bezier::polynomial (Axis a)const
 104 {
 105   Polynomial p (0.0);
 106   for (int j=0; j <= 3; j++)
 107     {
 108       p += control_[j][a]
 109         * Polynomial::power (j , Polynomial (0,1))*
 110         Polynomial::power (3 - j, Polynomial (1,-1))*
 111         binomial_coefficient(3, j);
 112     }
 113
 114   return p;
 115 }
 116
 117 /**
 118    Remove all numbers outside [0,1] from SOL
 119  */
 120 Array<Real>
 121 filter_solutions (Array<Real> sol)
 122 {
 123   for (int i = sol.size (); i--;)
 124     if (sol[i] < 0 || sol[i] >1)
 125       sol.del (i);
 126   return sol;
 127 }
 128
 129 /**
 130    find t such that derivative is proportional to DERIV
 131  */
 132 Array<Real>
 133 Bezier::solve_derivative (Offset deriv)const
 134 {
 135   Polynomial xp=polynomial (X_AXIS);
 136   Polynomial yp=polynomial (Y_AXIS);
 137   xp.differentiate ();
 138   yp.differentiate ();
 139
 140   Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
 141
 142   return filter_solutions (combine.solve ());
 143 }
 144
 145
 146 /*
 147   Find t such that curve_point (t)[AX] == COORDINATE
 148 */
 149 Array<Real>
 150 Bezier::solve_point (Axis ax, Real coordinate) const
 151 {
 152   Polynomial p(polynomial (ax));
 153   p.coefs_[0] -= coordinate;
 154
 155   Array<Real> sol (p.solve ());
 156   return filter_solutions (sol);
 157 }
 158
 159 /**
 160    Compute the bounding box dimensions in direction of A.
 161  */
 162 Interval
 163 Bezier::extent (Axis a)const
 164 {
 165   int o = (a+1)%NO_AXES;
 166   Offset d;
 167   d[Axis (o)] =1.0;
 168   Interval iv;
 169   Array<Real> sols (solve_derivative (d));
 170   sols.push (1.0);
 171   sols.push (0.0);
 172   for (int i= sols.size (); i--;)
 173     {
 174       Offset o (curve_point (sols[i]));
 175       iv.unite (Interval (o[a],o[a]));
 176     }
 177   return iv;
 178 }
 179
 180 /**
 181    Flip around axis A
 182  */
 183
 184 void
 185 Bezier::scale (Real x, Real y)
 186 {
 187   for (int i = CONTROL_COUNT; i--;)
 188     {
 189       control_[i][X_AXIS] = x * control_[i][X_AXIS];
 190       control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
 191     }
 192 }
 193
 194 void
 195 Bezier::rotate (Real phi)
 196 {
 197   Offset rot (complex_exp (Offset (0, phi)));
 198   for (int i = 0; i < CONTROL_COUNT; i++)
 199     control_[i] = complex_multiply (rot, control_[i]);
 200 }
 201
 202 void
 203 Bezier::translate (Offset o)
 204 {
 205   for (int i = 0; i < CONTROL_COUNT; i++)
 206     control_[i] += o;
 207 }
 208
 209 void
 210 Bezier::assert_sanity () const
 211 {
 212   for (int i=0; i < CONTROL_COUNT; i++)
 213     assert (!isnan (control_[i].length ())
 214             && !isinf (control_[i].length ()));
 215 }
 216
 217 void
 218 Bezier::reverse ()
 219 {
 220   Bezier b2;
 221   for (int i =0; i < CONTROL_COUNT; i++)
 222     b2.control_[CONTROL_COUNT-i-1] = control_[i];
 223   *this = b2;
 224 }