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++)
56 Formula of the bezier 3-spline
58 sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j) t^j
61 A is the axis of X coordinate.
65 Bezier::get_other_coordinate (Axis a, Real x) const
67 Axis other = Axis ((a +1)%NO_AXES);
68 Array<Real> ts = solve_point (a, x);
72 programming_error ("no solution found for Bezier intersection");
76 Offset c = curve_point (ts[0]);
78 if (fabs (c[a] - x) > 1e-8)
79 programming_error ("bezier intersection not correct?");
85 Bezier::curve_point (Real t) const
88 Real one_min_tj = (1 - t) * (1 - t) * (1 - t);
91 for (int j = 0; j < 4; j++)
93 o += control_[j] * binomial_coefficient (3, j)
94 * pow (t, j) * pow (1 - t, 3 - j);
98 one_min_tj /= (1 - t);
102 assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t)) < 1e-8);
103 assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t)) < 1e-8);
110 Bezier::polynomial (Axis a) const
113 for (int j = 0; j <= 3; j++)
116 += (control_[j][a] * binomial_coefficient (3, j))
117 * Polynomial::power (j, Polynomial (0, 1))
118 * Polynomial::power (3 - j, Polynomial (1, -1));
125 Remove all numbers outside [0, 1] from SOL
128 filter_solutions (Array<Real> sol)
130 for (int i = sol.size (); i--;)
131 if (sol[i] < 0 || sol[i] > 1)
137 find t such that derivative is proportional to DERIV
140 Bezier::solve_derivative (Offset deriv) const
142 Polynomial xp = polynomial (X_AXIS);
143 Polynomial yp = polynomial (Y_AXIS);
147 Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
149 return filter_solutions (combine.solve ());
153 Find t such that curve_point (t)[AX] == COORDINATE
156 Bezier::solve_point (Axis ax, Real coordinate) const
158 Polynomial p (polynomial (ax));
159 p.coefs_[0] -= coordinate;
161 Array<Real> sol (p.solve ());
162 return filter_solutions (sol);
166 Compute the bounding box dimensions in direction of A.
169 Bezier::extent (Axis a) const
171 int o = (a + 1)%NO_AXES;
175 Array<Real> sols (solve_derivative (d));
178 for (int i = sols.size (); i--;)
180 Offset o (curve_point (sols[i]));
181 iv.unite (Interval (o[a], o[a]));
190 Bezier::scale (Real x, Real y)
192 for (int i = CONTROL_COUNT; i--;)
194 control_[i][X_AXIS] = x * control_[i][X_AXIS];
195 control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
200 Bezier::rotate (Real phi)
202 Offset rot (complex_exp (Offset (0, phi)));
203 for (int i = 0; i < CONTROL_COUNT; i++)
204 control_[i] = complex_multiply (rot, control_[i]);
208 Bezier::translate (Offset o)
210 for (int i = 0; i < CONTROL_COUNT; i++)
215 Bezier::assert_sanity () const
217 for (int i = 0; i < CONTROL_COUNT; i++)
218 assert (!isnan (control_[i].length ())
219 && !isinf (control_[i].length ()));
226 for (int i = 0; i < CONTROL_COUNT; i++)
227 b2.control_[CONTROL_COUNT - i - 1] = control_[i];