2 bezier.cc -- implement Bezier and Bezier_bow
4 source file of the GNU LilyPond music typesetter
6 (c) 1998--2005 Jan Nieuwenhuizen <janneke@gnu.org>
13 #include "libc-extension.hh"
16 binomial_coefficient (Real over , int under)
22 x *= over / Real (under);
31 scale (Array<Offset>* array, Real x , Real y)
33 for (int i = 0; i < array->size (); i++)
35 (*array)[i][X_AXIS] = x* (*array)[i][X_AXIS];
36 (*array)[i][Y_AXIS] = y* (*array)[i][Y_AXIS];
41 rotate (Array<Offset>* array, Real phi)
43 Offset rot (complex_exp (Offset (0, phi)));
44 for (int i = 0; i < array->size (); i++)
45 (*array)[i] = complex_multiply (rot, (*array)[i]);
49 translate (Array<Offset>* array, Offset o)
51 for (int i = 0; i < array->size (); i++)
57 Formula of the bezier 3-spline
59 sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j) t^j
62 A is the axis of X coordinate.
66 Bezier::get_other_coordinate (Axis a, Real x) const
68 Axis other = Axis ((a +1)%NO_AXES);
69 Array<Real> ts = solve_point (a, x);
73 programming_error ("No solution found for Bezier intersection.");
77 Offset c = curve_point (ts[0]);
79 if (fabs (c[a] - x) > 1e-8)
80 programming_error ("Bezier intersection not correct?");
87 Bezier::curve_point (Real t)const
90 Real one_min_tj = (1-t)* (1-t)* (1-t);
93 for (int j = 0 ; j < 4; j++)
95 o += control_[j] * binomial_coefficient (3, j)
96 * pow (t, j) * pow (1-t, 3-j);
104 assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t))< 1e-8);
105 assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t))< 1e-8);
113 Bezier::polynomial (Axis a)const
116 for (int j = 0; j <= 3; j++)
119 (control_[j][a] * binomial_coefficient (3, j))
120 * Polynomial::power (j, Polynomial (0, 1))
121 * Polynomial::power (3 - j, Polynomial (1, -1));
128 Remove all numbers outside [0, 1] from SOL
131 filter_solutions (Array<Real> sol)
133 for (int i = sol.size (); i--;)
134 if (sol[i] < 0 || sol[i] >1)
140 find t such that derivative is proportional to DERIV
143 Bezier::solve_derivative (Offset deriv)const
145 Polynomial xp = polynomial (X_AXIS);
146 Polynomial yp = polynomial (Y_AXIS);
150 Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
152 return filter_solutions (combine.solve ());
157 Find t such that curve_point (t)[AX] == COORDINATE
160 Bezier::solve_point (Axis ax, Real coordinate) const
162 Polynomial p (polynomial (ax));
163 p.coefs_[0] -= coordinate;
165 Array<Real> sol (p.solve ());
166 return filter_solutions (sol);
170 Compute the bounding box dimensions in direction of A.
173 Bezier::extent (Axis a)const
175 int o = (a+1)%NO_AXES;
179 Array<Real> sols (solve_derivative (d));
182 for (int i = sols.size (); i--;)
184 Offset o (curve_point (sols[i]));
185 iv.unite (Interval (o[a], o[a]));
194 Bezier::scale (Real x, Real y)
196 for (int i = CONTROL_COUNT; i--;)
198 control_[i][X_AXIS] = x * control_[i][X_AXIS];
199 control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
204 Bezier::rotate (Real phi)
206 Offset rot (complex_exp (Offset (0, phi)));
207 for (int i = 0; i < CONTROL_COUNT; i++)
208 control_[i] = complex_multiply (rot, control_[i]);
212 Bezier::translate (Offset o)
214 for (int i = 0; i < CONTROL_COUNT; i++)
219 Bezier::assert_sanity () const
221 for (int i = 0; i < CONTROL_COUNT; i++)
222 assert (!isnan (control_[i].length ())
223 && !isinf (control_[i].length ()));
230 for (int i = 0; i < CONTROL_COUNT; i++)
231 b2.control_[CONTROL_COUNT-i-1] = control_[i];