/*
- 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 Jan Nieuwenhuizen <jan@digicash.com>
+ 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.
+
+ You should have received a copy of the GNU General Public License
+ along with LilyPond. If not, see <http://www.gnu.org/licenses/>.
*/
-#include <math.h>
-#include "offset.hh"
#include "bezier.hh"
-#include "dimen.hh"
-#include "paper-def.hh"
+#include "warn.hh"
+#include "libc-extension.hh"
+
+Real binomial_coefficient_3[]
+=
+{
+ 1, 3, 3, 1
+};
-Bezier::Bezier (int steps_i)
+void
+scale (vector<Offset> *array, Real x, Real y)
{
- steps_i_ = steps_i;
- curve_ = new Offset [steps_i_ + 1];
+ 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];
+ }
}
-Bezier::~Bezier ()
+void
+rotate (vector<Offset> *array, Real deg)
{
- delete[] curve_;
+ Offset rot (offset_directed (deg));
+ for (vsize i = 0; i < array->size (); i++)
+ (*array)[i] = complex_multiply (rot, (*array)[i]);
}
-//from GNU gs3.33: ega.c
void
-Bezier::calc (Offset control[4])
-{
- Real dt = 1.0 / steps_i_;
- Real cx = 3.0 * (control[1].x() - control[0].x());
- Real bx = 3.0 * (control[2].x() - control[1].x()) - cx;
- Real ax = control[3].x() - (control[0].x() + cx + bx);
- Real cy = 3.0 * (control[1].y () - control[0].y ());
- Real by = 3.0 * (control[2].y () - control[1].y ()) - cy;
- Real ay = control[3].y () - (control[0].y () + cy + by);
- Real t = 0.0;
- int i = 0;
- while ( t <= 1.0 )
- {
- curve_[i].x() = ((ax * t + bx) * t + cx) * t + control[0].x();
- curve_[i++].y () = ((ay * t + by) * t + cy) * t + control[0].y ();
- t += dt;
+translate (vector<Offset> *array, Offset o)
+{
+ 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
+
+
+ A is the axis of X coordinate.
+*/
+
+Real
+Bezier::get_other_coordinate (Axis a, Real x) const
+{
+ 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");
+ return 0.0;
}
+
+#ifdef PARANOID
+ Offset c = curve_point (ts[0]);
+ if (fabs (c[a] - x) > 1e-8)
+ 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::y (Real x)
+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
{
- if (x <= curve_[0].x())
- return curve_[0].y ();
- for (int i = 1; i < steps_i_; i++ )
+ 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);
+
+ Offset o;
+ for (int j = 0; j < 4; j++)
{
- if (x < curve_[i].x())
- {
- Real lin = (x - curve_[i-1].x()) / (curve_[i].x() - curve_[i-1].x());
- return curve_[i-1].y () + lin * (curve_[i].y () - curve_[i-1].y ());
- }
+ o += control_[j] * binomial_coefficient_3[j]
+ * tj * one_min_tj[3 - j];
+
+ tj *= t;
}
- return curve_[steps_i_-1].y ();
+
+#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);
+#endif
+
+ return o;
+}
+
+// The return value is normalized unless zero or indefinite.
+Offset
+Bezier::dir_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];
+
+ return (third_order[1] - third_order[0]).direction ();
}
+/*
+ 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
+{
+ Polynomial p (0.0);
+ Polynomial q;
+ for (int j = 0; j <= 3; j++)
+ {
+ q = poly_cache.terms_[j];
+ q *= control_[j][a];
+ p += q;
+ }
+
+ return p;
+}
+
+/**
+ 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
+*/
+vector<Real>
+Bezier::solve_derivative (Offset deriv) const
+{
+ 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 ());
+}
-Bezier_bow::Bezier_bow (Paper_def* paper_l)
- : Bezier(10)
+/*
+ Find t such that curve_point (t)[AX] == COORDINATE
+*/
+vector<Real>
+Bezier::solve_point (Axis ax, Real coordinate) const
{
- paper_l_ = paper_l;
+ Polynomial p (polynomial (ax));
+ p.coefs_[0] -= coordinate;
+
+ 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);
-/*
- from feta-sleur.ly
+ // 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)
- slurheightlimit#:=staffsize#/2;
- sluralpha:=slurheightlimit#*pi/2;
- % slurratio:=1/3;
- slurratio:=0.3333;
- slurbeta:=3/4*pi*slurratio/sluralpha;
+ 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]);
+ }
- b#:=length(dx#,dy#);
- % ugh: huh? 2/5
- indent#:=2/5*sluralpha*atan(slurbeta*b#);
- height:=(indent+h)*d;
- z1=(0,0);
- z2=(b,0);
- z3=(indent,height);
- z4=(b-indent,height);
+ // 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]);
+ }
- boogje:=boogje rotated angle(dxs,dys);
+ 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
+{
+ int o = (a + 1) % NO_AXES;
+ Offset d;
+ d[Axis (o)] = 1.0;
+ Interval iv;
+ 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]));
+ }
+ 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_bow::calc (Real dx, Real dy, Real h, Real d)
-{
- // ugh
- Real pi = M_PI;
- // ugh
- Real staffsize_f = paper_l_->get_var ("barsize");
- Real height_limit = staffsize_f;
- Real alpha = height_limit * pi / 2.0;
- Real ratio = 1.0/3.0;
- Real beta = 3.0/4.0 * pi * ratio/alpha;
-
- Real b = sqrt (dx * dx + dy * dy);
- Real indent = 2.0/5.0 * alpha * atan (beta * b);
- // ugh, ugly height hack, see lily-ps-defs.tex
- Real height = (indent + h) * d;
-
- Offset control[4];
- control[0] = Offset(0, 0);
- control[1] = Offset(indent, height);
- control[2] = Offset(b - indent, height);
- control[3] = Offset( b, 0 );
-
- Real phi = dx ? atan (dy/dx) : sign (dy) * pi / 2.0;
- Real sphi = sin (phi);
- Real cphi = cos (phi);
- for (int i = 1; i < 4; i++)
+Bezier::scale (Real x, Real y)
+{
+ for (int i = CONTROL_COUNT; i--;)
+ {
+ control_[i][X_AXIS] = x * control_[i][X_AXIS];
+ control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
+ }
+}
+
+void
+Bezier::rotate (Real deg)
+{
+ Offset rot (offset_directed (deg));
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ control_[i] = complex_multiply (rot, control_[i]);
+}
+
+void
+Bezier::translate (Offset o)
+{
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ control_[i] += o;
+}
+
+void
+Bezier::assert_sanity () const
+{
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ assert (!isnan (control_[i].length ())
+ && !isinf (control_[i].length ()));
+}
+
+void
+Bezier::reverse ()
+{
+ Bezier b2;
+ for (int i = 0; i < CONTROL_COUNT; 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++)
{
- control[i].x() = cphi * control[i].x() - sphi * control[i].y ();
- control[i].y () = sphi * control[i].x() + cphi * control[i].y ();
+ left_part->control_[i] = p[0][CONTROL_COUNT - 1 - i];
+ right_part->control_[i] = p[i][i];
}
- Bezier::calc (control);
}
+/*
+ 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