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"
15 Real binomial_coefficient_3[] = {1,3 ,3, 1};
18 binomial_coefficient (Real over, int under)
24 x *= over / Real (under);
33 scale (Array<Offset> *array, Real x, Real y)
35 for (int i = 0; i < array->size (); i++)
37 (*array)[i][X_AXIS] = x * (*array)[i][X_AXIS];
38 (*array)[i][Y_AXIS] = y * (*array)[i][Y_AXIS];
43 rotate (Array<Offset> *array, Real phi)
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]);
51 translate (Array<Offset> *array, Offset o)
53 for (int i = 0; i < array->size (); i++)
58 Formula of the bezier 3-spline
60 sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j) t^j
63 A is the axis of X coordinate.
67 Bezier::get_other_coordinate (Axis a, Real x) const
69 Axis other = Axis ((a +1)%NO_AXES);
70 Array<Real> ts = solve_point (a, x);
74 programming_error ("no solution found for Bezier intersection");
78 Offset c = curve_point (ts[0]);
80 if (fabs (c[a] - x) > 1e-8)
81 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);
100 one_min_tj /= (1 - t);
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);
112 Bezier::polynomial (Axis a) const
115 for (int j = 0; j <= 3; j++)
118 += (control_[j][a] * binomial_coefficient_3[j])
119 * Polynomial::power (j, Polynomial (0, 1))
120 * Polynomial::power (3 - j, Polynomial (1, -1));
127 Remove all numbers outside [0, 1] from SOL
130 filter_solutions (Array<Real> sol)
132 for (int i = sol.size (); i--;)
133 if (sol[i] < 0 || sol[i] > 1)
139 find t such that derivative is proportional to DERIV
142 Bezier::solve_derivative (Offset deriv) const
144 Polynomial xp = polynomial (X_AXIS);
145 Polynomial yp = polynomial (Y_AXIS);
149 Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
151 return filter_solutions (combine.solve ());
155 Find t such that curve_point (t)[AX] == COORDINATE
158 Bezier::solve_point (Axis ax, Real coordinate) const
160 Polynomial p (polynomial (ax));
161 p.coefs_[0] -= coordinate;
163 Array<Real> sol (p.solve ());
164 return filter_solutions (sol);
168 Compute the bounding box dimensions in direction of A.
171 Bezier::extent (Axis a) const
173 int o = (a + 1)%NO_AXES;
177 Array<Real> sols (solve_derivative (d));
180 for (int i = sols.size (); i--;)
182 Offset o (curve_point (sols[i]));
183 iv.unite (Interval (o[a], o[a]));
192 Bezier::scale (Real x, Real y)
194 for (int i = CONTROL_COUNT; i--;)
196 control_[i][X_AXIS] = x * control_[i][X_AXIS];
197 control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
202 Bezier::rotate (Real phi)
204 Offset rot (complex_exp (Offset (0, phi)));
205 for (int i = 0; i < CONTROL_COUNT; i++)
206 control_[i] = complex_multiply (rot, control_[i]);
210 Bezier::translate (Offset o)
212 for (int i = 0; i < CONTROL_COUNT; i++)
217 Bezier::assert_sanity () const
219 for (int i = 0; i < CONTROL_COUNT; i++)
220 assert (!isnan (control_[i].length ())
221 && !isinf (control_[i].length ()));
228 for (int i = 0; i < CONTROL_COUNT; i++)
229 b2.control_[CONTROL_COUNT - i - 1] = control_[i];