/*
- bezier.cc -- implement Bezier and Bezier_bow
+ This file is part of LilyPond, the GNU music typesetter.
- source file of the GNU LilyPond music typesetter
+ Copyright (C) 1998--2015 Jan Nieuwenhuizen <janneke@gnu.org>
- (c) 1998--2004 Jan Nieuwenhuizen <janneke@gnu.org>
-*/
+ LilyPond is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+ LilyPond is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
-#include <math.h>
+ You should have received a copy of the GNU General Public License
+ along with LilyPond. If not, see <http://www.gnu.org/licenses/>.
+*/
#include "bezier.hh"
#include "warn.hh"
#include "libc-extension.hh"
-Real
-binomial_coefficient (Real over , int under)
+Real binomial_coefficient_3[]
+=
{
- Real x = 1.0;
-
- while (under)
- {
- x *= over / Real (under);
-
- over -= 1.0;
- under --;
- }
- return x;
-}
+ 1, 3, 3, 1
+};
void
-scale (Array<Offset>* array, Real x , Real y)
+scale (vector<Offset> *array, Real x, Real y)
{
- for (int i = 0; i < array->size (); i++)
+ for (vsize i = 0; i < array->size (); i++)
{
- (*array)[i][X_AXIS] = x* (*array)[i][X_AXIS];
- (*array)[i][Y_AXIS] = y* (*array)[i][Y_AXIS];
+ (*array)[i][X_AXIS] = x * (*array)[i][X_AXIS];
+ (*array)[i][Y_AXIS] = y * (*array)[i][Y_AXIS];
}
}
void
-rotate (Array<Offset>* array, Real phi)
+rotate (vector<Offset> *array, Real phi)
{
Offset rot (complex_exp (Offset (0, phi)));
- for (int i = 0; i < array->size (); i++)
+ for (vsize i = 0; i < array->size (); i++)
(*array)[i] = complex_multiply (rot, (*array)[i]);
}
void
-translate (Array<Offset>* array, Offset o)
+translate (vector<Offset> *array, Offset o)
{
- for (int i = 0; i < array->size (); i++)
+ for (vsize i = 0; i < array->size (); i++)
(*array)[i] += o;
}
/*
-
Formula of the bezier 3-spline
- sum_{j= 0}^3 (3 over j) z_j (1-t)^ (3-j) t^j
+ sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j) t^j
A is the axis of X coordinate.
- */
+*/
Real
-Bezier::get_other_coordinate (Axis a, Real x) const
+Bezier::get_other_coordinate (Axis a, Real x) const
{
- Axis other = Axis ((a +1)%NO_AXES);
- Array<Real> ts = solve_point (a, x);
+ Axis other = Axis ((a + 1) % NO_AXES);
+ vector<Real> ts = solve_point (a, x);
if (ts.size () == 0)
{
- programming_error ("No solution found for Bezier intersection.");
+ programming_error ("no solution found for Bezier intersection");
return 0.0;
}
-
- Offset c = curve_point (ts[0]);
+#ifdef PARANOID
+ Offset c = curve_point (ts[0]);
if (fabs (c[a] - x) > 1e-8)
- programming_error ("Bezier intersection not correct?");
-
- return c[other];
+ programming_error ("bezier intersection not correct?");
+#endif
+
+ return curve_coordinate (ts[0], other);
+}
+
+vector<Real>
+Bezier::get_other_coordinates (Axis a, Real x) const
+{
+ Axis other = other_axis (a);
+ vector<Real> ts = solve_point (a, x);
+ vector<Real> sols;
+ for (vsize i = 0; i < ts.size (); i++)
+ sols.push_back (curve_coordinate (ts[i], other));
+ return sols;
}
+Real
+Bezier::curve_coordinate (Real t, Axis a) const
+{
+ Real tj = 1;
+ Real one_min_tj[4];
+ one_min_tj[0] = 1;
+ for (int i = 1; i < 4; i++)
+ one_min_tj[i] = one_min_tj[i - 1] * (1 - t);
+
+ Real r = 0.0;
+ for (int j = 0; j < 4; j++)
+ {
+ r += control_[j][a] * binomial_coefficient_3[j]
+ * tj * one_min_tj[3 - j];
+
+ tj *= t;
+ }
+
+ return r;
+}
Offset
-Bezier::curve_point (Real t)const
+Bezier::curve_point (Real t) const
{
Real tj = 1;
- Real one_min_tj = (1-t)* (1-t)* (1-t);
+ Real one_min_tj[4];
+ one_min_tj[0] = 1;
+ for (int i = 1; i < 4; i++)
+ one_min_tj[i] = one_min_tj[i - 1] * (1 - t);
Offset o;
- for (int j= 0 ; j < 4; j++)
+ for (int j = 0; j < 4; j++)
{
- o += control_[j] * binomial_coefficient (3, j)
- * pow (t,j) * pow (1-t, 3-j);
+ o += control_[j] * binomial_coefficient_3[j]
+ * tj * one_min_tj[3 - j];
tj *= t;
- if (1-t)
- one_min_tj /= (1-t);
}
#ifdef PARANOID
- assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t))< 1e-8);
- assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t))< 1e-8);
+ assert (fabs (o[X_AXIS] - polynomial (X_AXIS).eval (t)) < 1e-8);
+ assert (fabs (o[Y_AXIS] - polynomial (Y_AXIS).eval (t)) < 1e-8);
#endif
-
+
return o;
}
+Real
+Bezier::slope_at_point (Real t) const
+{
+ Offset second_order[3];
+ Offset third_order[2];
+
+ for (vsize i = 0; i < 3; i++)
+ second_order[i] = ((control_[i + 1] - control_[i]) * t) + control_[i];
+
+ for (vsize i = 0; i < 2; i++)
+ third_order[i] = ((second_order[i + 1] - second_order[i]) * t) + second_order[i];
+
+ if (third_order[1][X_AXIS] - third_order[0][X_AXIS] == 0)
+ return infinity_f;
+
+ return (third_order[1][Y_AXIS] - third_order[0][Y_AXIS]) / (third_order[1][X_AXIS] - third_order[0][X_AXIS]);
+}
+
+/*
+ Cache binom (3, j) t^j (1-t)^{3-j}
+*/
+struct Polynomial_cache
+{
+ Polynomial terms_[4];
+ Polynomial_cache ()
+ {
+ for (int j = 0; j <= 3; j++)
+ terms_[j]
+ = binomial_coefficient_3[j]
+ * Polynomial::power (j, Polynomial (0, 1))
+ * Polynomial::power (3 - j, Polynomial (1, -1));
+ }
+};
+
+static Polynomial_cache poly_cache;
Polynomial
-Bezier::polynomial (Axis a)const
+Bezier::polynomial (Axis a) const
{
Polynomial p (0.0);
- for (int j= 0; j <= 3; j++)
+ Polynomial q;
+ for (int j = 0; j <= 3; j++)
{
- p +=
- (control_[j][a] * binomial_coefficient (3, j))
- * Polynomial::power (j, Polynomial (0, 1))
- * Polynomial::power (3 - j, Polynomial (1, -1));
+ q = poly_cache.terms_[j];
+ q *= control_[j][a];
+ p += q;
}
return p;
}
/**
- Remove all numbers outside [0,1] from SOL
- */
-Array<Real>
-filter_solutions (Array<Real> sol)
-{
- for (int i = sol.size (); i--;)
- if (sol[i] < 0 || sol[i] >1)
- sol.del (i);
+ Remove all numbers outside [0, 1] from SOL
+*/
+vector<Real>
+filter_solutions (vector<Real> sol)
+{
+ for (vsize i = sol.size (); i--;)
+ if (sol[i] < 0 || sol[i] > 1)
+ sol.erase (sol.begin () + i);
return sol;
}
/**
find t such that derivative is proportional to DERIV
- */
-Array<Real>
-Bezier::solve_derivative (Offset deriv)const
+*/
+vector<Real>
+Bezier::solve_derivative (Offset deriv) const
{
- Polynomial xp=polynomial (X_AXIS);
- Polynomial yp=polynomial (Y_AXIS);
+ Polynomial xp = polynomial (X_AXIS);
+ Polynomial yp = polynomial (Y_AXIS);
xp.differentiate ();
yp.differentiate ();
-
+
Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
return filter_solutions (combine.solve ());
}
-
/*
Find t such that curve_point (t)[AX] == COORDINATE
*/
-Array<Real>
+vector<Real>
Bezier::solve_point (Axis ax, Real coordinate) const
{
Polynomial p (polynomial (ax));
p.coefs_[0] -= coordinate;
-
- Array<Real> sol (p.solve ());
+
+ vector<Real> sol (p.solve ());
return filter_solutions (sol);
}
+/**
+ For the portion of the curve between L and R along axis AX,
+ return the bounding box limit in direction D along the cross axis to AX.
+ If there is no portion between L and R, return 0.0 and report error.
+*/
+Real
+Bezier::minmax (Axis ax, Real l, Real r, Direction d) const
+{
+ Axis bx = other_axis (ax);
+
+ // The curve could hit its bounding box limit along BX at:
+ // points where the curve is parallel to AX,
+ Offset vec (0.0, 0.0);
+ vec[ax] = 1.0;
+ vector<Real> sols (solve_derivative (vec));
+ // or endpoints of the curve,
+ sols.push_back (0.999);
+ sols.push_back (0.001);
+ // (using points just inside the ends, so that an endpoint is evaulated
+ // if it falls within rounding error of L or R and the curve lies inside)
+
+ Interval iv;
+ for (vsize i = sols.size (); i--;)
+ {
+ Offset p (curve_point (sols[i]));
+ if (p[ax] >= l && p[ax] <= r)
+ iv.add_point (p[bx]);
+ }
+
+ // or intersections of the curve with the bounding lines at L and R.
+ Interval lr (l, r);
+ for (LEFT_and_RIGHT (dir))
+ {
+ vector<Real> v = get_other_coordinates (ax, lr[dir]);
+ for (vsize i = v.size (); i--;)
+ iv.add_point (v[i]);
+ }
+
+ if (iv.is_empty ())
+ {
+ programming_error ("Bezier curve does not cross region of concern");
+ return 0.0;
+ }
+
+ return iv.at (d);
+}
+
/**
Compute the bounding box dimensions in direction of A.
- */
+*/
Interval
-Bezier::extent (Axis a)const
+Bezier::extent (Axis a) const
{
- int o = (a+1)%NO_AXES;
+ int o = (a + 1) % NO_AXES;
Offset d;
- d[Axis (o)] =1.0;
+ d[Axis (o)] = 1.0;
Interval iv;
- Array<Real> sols (solve_derivative (d));
- sols.push (1.0);
- sols.push (0.0);
- for (int i= sols.size (); i--;)
+ vector<Real> sols (solve_derivative (d));
+ sols.push_back (1.0);
+ sols.push_back (0.0);
+ for (vsize i = sols.size (); i--;)
{
Offset o (curve_point (sols[i]));
- iv.unite (Interval (o[a],o[a]));
+ iv.unite (Interval (o[a], o[a]));
}
return iv;
}
+Interval
+Bezier::control_point_extent (Axis a) const
+{
+ Interval ext;
+ for (int i = CONTROL_COUNT; i--;)
+ ext.add_point (control_[i][a]);
+
+ return ext;
+}
+
/**
Flip around axis A
- */
+*/
void
Bezier::scale (Real x, Real y)
{
void
Bezier::assert_sanity () const
{
- for (int i= 0; i < CONTROL_COUNT; i++)
+ for (int i = 0; i < CONTROL_COUNT; i++)
assert (!isnan (control_[i].length ())
- && !isinf (control_[i].length ()));
+ && !isinf (control_[i].length ()));
}
void
{
Bezier b2;
for (int i = 0; i < CONTROL_COUNT; i++)
- b2.control_[CONTROL_COUNT-i-1] = control_[i];
+ b2.control_[CONTROL_COUNT - i - 1] = control_[i];
*this = b2;
}
+
+/*
+ Subdivide a bezier at T into LEFT_PART and RIGHT_PART
+ using deCasteljau's algorithm.
+*/
+void
+Bezier::subdivide (Real t, Bezier *left_part, Bezier *right_part) const
+{
+ Offset p[CONTROL_COUNT][CONTROL_COUNT];
+
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ p[i][CONTROL_COUNT - 1 ] = control_[i];
+ for (int j = CONTROL_COUNT - 2; j >= 0; j--)
+ for (int i = 0; i < CONTROL_COUNT - 1; i++)
+ p[i][j] = p[i][j + 1] + t * (p[i + 1][j + 1] - p[i][j + 1]);
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ {
+ left_part->control_[i] = p[0][CONTROL_COUNT - 1 - i];
+ right_part->control_[i] = p[i][i];
+ }
+}
+
+/*
+ Extract a portion of a bezier from T_MIN to T_MAX
+*/
+
+Bezier
+Bezier::extract (Real t_min, Real t_max) const
+{
+ if ((t_min < 0) || (t_max) > 1)
+ programming_error
+ ("bezier extract arguments outside of limits: curve may have bad shape");
+ if (t_min >= t_max)
+ programming_error
+ ("lower bezier extract value not less than upper value: curve may have bad shape");
+ Bezier bez1, bez2, bez3, bez4;
+ if (t_min == 0.0)
+ bez2 = *this;
+ else
+ subdivide (t_min, &bez1, &bez2);
+ if (t_max == 1.0)
+ return bez2;
+ else
+ {
+ bez2.subdivide ((t_max - t_min) / (1 - t_min), &bez3, &bez4);
+ return bez3;
+ }
+}
projects / lilypond.git / blobdiff