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