source file of the GNU LilyPond music typesetter
- (c) 1998 Jan Nieuwenhuizen <jan@digicash.com>
+ (c) 1998--2008 Jan Nieuwenhuizen <janneke@gnu.org>
*/
-#include <math.h>
#include "bezier.hh"
-#include "direction.hh"
-
-#ifndef STANDALONE
-#include "direction.hh"
-#include "dimen.hh"
-#include "paper-def.hh"
-#include "debug.hh"
-#include "main.hh"
-#define SLUR_DOUT if (check_debug && !monitor->silent_b ("Slur")) cout
-#else
-#define SLUR_DOUT cerr
-#endif
+#include "warn.hh"
+#include "libc-extension.hh"
-void
-Curve::flipy ()
-{
- // ugh, Offset should have mirror funcs
- for (int i = 0; i < size (); i++)
- (*this)[i].mirror (Y_AXIS);
-}
+Real binomial_coefficient_3[] = {
+ 1, 3, 3, 1
+};
-int
-Curve::largest_disturbing ()
+void
+scale (vector<Offset> *array, Real x, Real y)
{
- Real alpha = 0;
- int j = 0;
- for (int i = 1; i < size (); i++)
+ for (vsize i = 0; i < array->size (); i++)
{
- if ((*this)[i].y () > 0)
- {
- Real phi = (*this)[i].y () / (*this)[i].x ();
- if (phi > alpha)
- {
- alpha = phi;
- j = i;
- }
- }
+ (*array)[i][X_AXIS] = x * (*array)[i][X_AXIS];
+ (*array)[i][Y_AXIS] = y * (*array)[i][Y_AXIS];
}
- return j;
}
void
-Curve::rotate (Real phi)
+rotate (vector<Offset> *array, Real phi)
{
- Offset rot (complex_exp (Offset (0,phi)));
-
- for (int i = 1; i < size (); i++)
- (*this)[i] = complex_multiply (rot, (*this)[i]);
+ Offset rot (complex_exp (Offset (0, phi)));
+ for (vsize i = 0; i < array->size (); i++)
+ (*array)[i] = complex_multiply (rot, (*array)[i]);
}
void
-Curve::translate (Offset o)
+translate (vector<Offset> *array, Offset o)
{
- for (int i = 1; i < size (); i++)
- (*this)[i] += o;
+ for (vsize i = 0; i < array->size (); i++)
+ (*array)[i] += o;
}
-Bezier::Bezier (int steps)
-{
- control_.set_size (4);
- curve_.set_size (steps);
-}
+/*
+ Formula of the bezier 3-spline
-//from GNU gs3.33: ega.c
-void
-Bezier::calc ()
-{
- Real dt = 1.0 / curve_.size ();
- Offset c = 3.0 * (control_[1] - control_[0]);
- Offset b = 3.0 * (control_[2] - control_[1]) - c;
- Offset a = control_[3] - (control_[0] + c + b);
- Real t = 0.0;
- for (int i = 0; i < curve_.size (); i++ )
- {
- curve_[i] = ((a * t + b) * t + c) * t + control_[0];
- t += dt;
- }
-}
+ sum_{j = 0}^3 (3 over j) z_j (1-t)^ (3-j) t^j
-void
-Bezier::set (Array<Offset> points)
-{
- assert (points.size () == 4);
- control_ = points;
-}
+
+ A is the axis of X coordinate.
+*/
Real
-Bezier::y (Real x)
+Bezier::get_other_coordinate (Axis a, Real x) const
{
- if (x <= curve_[0].x ())
- return curve_[0].y ();
- for (int i = 1; i < curve_.size (); i++ )
+ Axis other = Axis ((a +1) % NO_AXES);
+ vector<Real> ts = solve_point (a, x);
+
+ if (ts.size () == 0)
{
- if (x < curve_[i].x ())
- // ^ ? see below
- {
- Real lin = (x - curve_[i-1].x ()) / (curve_[i] - curve_[i-1]).x ();
- // ^ ?
- return (curve_[i-1] + lin * (curve_[i] - curve_[i-1])).y ();
- }
+ programming_error ("no solution found for Bezier intersection");
+ return 0.0;
}
- return curve_[curve_.size ()-1].y ();
-}
+#ifdef PARANOID
+ Offset c = curve_point (ts[0]);
+ if (fabs (c[a] - x) > 1e-8)
+ programming_error ("bezier intersection not correct?");
+#endif
-Bezier_bow::Bezier_bow (Paper_def* paper_l)
- : Bezier(10)
-{
- paper_l_ = paper_l;
- return_.set_size (4);
+ return curve_coordinate (ts[0], other);
}
-/*
- from feta-sleur.ly
-
- slurheightlimit#:=staffsize#/2;
- sluralpha:=slurheightlimit#*pi/2;
- % slurratio:=1/3;
- slurratio:=0.3333;
- slurbeta:=3/4*pi*slurratio/sluralpha;
-
- 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);
-
- boogje:=boogje rotated angle(dxs,dys);
-*/
-
-void
-Bezier_bow::blow_fit ()
+Real
+Bezier::curve_coordinate (Real t, Axis a) const
{
- Real dy1 = check_fit_f ();
- if (!dy1)
- return;
-
- // be careful not to take too big step
- Real f = 0.75;
- Real h1 = dy1 * f;
- control_[1].y () += h1;
- control_[2].y () += h1;
- return_[1].y () += h1;
- return_[2].y () += h1;
-
- Real dy2 = check_fit_f ();
- if (!dy2)
- return;
-
-#ifndef STANDALONE
- Real epsilon = paper_l_->rule_thickness ();
-#else
- Real epsilon = 1.5 * 0.4 PT;
-#endif
- if (abs (dy2 - dy1) < epsilon)
- return;
-
- /*
- Assume
- dy = B (h)
- with
- B (h) = a * h + b;
-
- Then we get for h : B (h) = 0
+ 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];
- B(0) = dy1 = a * 0 + b => b = dy1
- B(h1) = dy2 = a * h1 + b => a * f * dy1 + b = dy2
+ tj *= t;
+ }
- =>
+ return r;
+}
- a * dy1 / 2 + dy1 = dy2 => a = (dy2 - dy1) / (f * dy1)
- */
+Offset
+Bezier::curve_point (Real t) 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);
+
+ Offset o;
+ for (int j = 0; j < 4; j++)
+ {
+ o += control_[j] * binomial_coefficient_3[j]
+ * tj * one_min_tj[3 - j];
- Real a = (dy2 - dy1) / (f * dy1);
- Real b = dy1;
- Real h = -b / a;
+ tj *= t;
+ }
- control_[1].y () += -h1 +h;
- control_[2].y () = -h1 +h;
- return_[1].y () = -h1 +h;
- return_[2].y () = -h1 +h;
-}
+#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
-Real
-Bezier_bow::calc_f (Real height)
-{
- transform ();
- calc_default (height);
- Bezier::calc ();
-
- Real dy = check_fit_f ();
- calc_return (0, 0);
-
- transform_controls_back ();
- return dy;
+ return o;
}
-void
-Bezier_bow::calc ()
+/*
+ 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
{
- transform ();
- calc_default (0);
- Bezier::calc ();
-
- if (check_fit_bo ())
- calc_return (0, 0);
- else
+ Polynomial p (0.0);
+ Polynomial q;
+ for (int j = 0; j <= 3; j++)
{
- calc_controls ();
- blow_fit ();
+ q = poly_cache.terms_[j];
+ q *= control_[j][a];
+ p += q;
}
- transform_controls_back ();
+ return p;
}
-void
-Bezier_bow::calc_return (Real begin_alpha, Real end_alpha)
+/**
+ Remove all numbers outside [0, 1] from SOL
+*/
+vector<Real>
+filter_solutions (vector<Real> sol)
{
-#ifndef STANDALONE
- Real thick = 1.8 * paper_l_->rule_thickness ();
-#else
- Real thick = 10.0 * 1.8 * 0.4 PT;
-#endif
- return_[0] = control_[3];
-
- return_[1] = control_[2] - thick * complex_exp (Offset (0, 90 + end_alpha));
- return_[2] = control_[1] - thick * complex_exp (Offset (0, 90 - begin_alpha));
-
- /*
- return_[1].x () = control_[2].x () - thick * cos (90 + end_alpha);
- return_[1].y () = control_[2].y () - thick * sin (90 + end_alpha);
- return_[2].x () = control_[1].x () - thick * cos (90 - begin_alpha);
- return_[2].y () = control_[1].y () - thick * sin (90 - begin_alpha);
- */
- return_[3] = control_[0];
+ for (vsize i = sol.size (); i--;)
+ if (sol[i] < 0 || sol[i] > 1)
+ sol.erase (sol.begin () + i);
+ return sol;
}
-void
-Bezier_bow::calc_controls ()
+/**
+ find t such that derivative is proportional to DERIV
+*/
+vector<Real>
+Bezier::solve_derivative (Offset deriv) const
{
- // ugh: tooo steep
-// Real default_rc = atan (control_[1].y () / control_[1].x ());
-
- Offset ijk_p (control_[3].x () / 2, control_[1].y ());
- SLUR_DOUT << "ijk: " << ijk_p.x () << ", " << ijk_p.y () << endl;
-
- Real default_rc = ijk_p.y () / ijk_p.x ();
-
- int begin_disturb = encompass_.largest_disturbing ();
- Offset begin_p = begin_disturb ? Offset (encompass_[begin_disturb].x (),
- encompass_[begin_disturb].y ()) : ijk_p;
- Real begin_rc = begin_p.y () / begin_p.x ();
- if (default_rc > begin_rc)
- {
- begin_p = ijk_p;
- begin_rc = default_rc;
- }
+ Polynomial xp = polynomial (X_AXIS);
+ Polynomial yp = polynomial (Y_AXIS);
+ xp.differentiate ();
+ yp.differentiate ();
- // ugh
- Curve reversed;
- reversed.set_size (encompass_.size ());
- Real b = control_[3].x ();
- for (int i = 0; i < encompass_.size (); i++ )
- {
- reversed[i] = Offset (b,0) - encompass_[encompass_.size () - i -1];
- /*
- reversed[i].x () = b - encompass_[encompass_.size () - i - 1].x ();
- reversed[i].y () = encompass_[encompass_.size () - i - 1].y ();
- */
- }
+ Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
- int end_disturb = reversed.largest_disturbing ();
- end_disturb = end_disturb ? encompass_.size () - end_disturb - 1 : 0;
- Offset end_p = end_disturb ? Offset (encompass_[end_disturb].x (),
- encompass_[end_disturb].y ()) : ijk_p;
- Real end_rc = end_p.y () / (control_[3].x () - end_p.x ());
- if (default_rc > end_rc)
- {
- end_p = ijk_p;
- end_rc = default_rc;
- }
- SLUR_DOUT << "begin " << begin_p.x () << ", " << begin_p.y () << endl;
- SLUR_DOUT << "end " << end_p.x () << ", " << end_p.y () << endl;
-
- Real height =control_[1].y ();
- for (int i = 0; i < encompass_.size (); i++ )
- height = height >? encompass_[i].y ();
-
- // emperic computer science:
- // * tangents somewhat steeper than minimal line
- Real rc_correct = 2.4;
-
- begin_rc *= rc_correct;
- end_rc *= rc_correct;
- Real rc1 = begin_rc;
- Real rc2 = -end_rc;
-
- Real begin_alpha = atan (begin_rc);
- Real end_alpha = atan (-end_rc);
- Real theta = (begin_alpha - end_alpha) / 2;
-
- // if we have two disturbing points, have height line through those...
- /*
- UGH UGH UGH! NEVER compare floats with ==
- */
- if (!((begin_p.x () == end_p.x ()) && (begin_p.y () == end_p.y ())))
- theta = atan (end_p.y () - begin_p.y ()) / (end_p.x () - begin_p.x ());
-
- Real rc3 = tan (theta);
- // ugh: be less steep
- rc3 /= 2*rc_correct;
-
- Real c2 = -rc2 * control_[3].x ();
- Real c3 = begin_p.y () > end_p.y () ? begin_p.y ()
- - rc3 * begin_p.x () : end_p.y () - rc3 * end_p.x ();
-
- SLUR_DOUT << "y1 = " << rc1 << " x + 0" << endl;
- SLUR_DOUT << "y2 = " << rc2 << " x + " << c2 << endl;
- SLUR_DOUT << "y3 = " << rc3 << " x + " << c3 << endl;
- control_[1].x () = c3 / (rc1 - rc3);
- control_[1].y () = rc1 * control_[1].x ();
- control_[2].x () = (c3 - c2) / (rc2 - rc3);
- SLUR_DOUT << "c2.x () = " << control_[2].x () << endl;
- SLUR_DOUT << "(c3 - c2) = " << (c3 - c2) << endl;
- SLUR_DOUT << "(rc2 - rc3) = " << (rc2 - rc3) << endl;
- control_[2].y () = rc2 * control_[2].x () + c2;
- SLUR_DOUT << "c2.y ()" << control_[2].y () << endl;
-
- calc_return (begin_alpha, end_alpha);
+ return filter_solutions (combine.solve ());
}
-bool
-Bezier_bow::check_fit_bo ()
+/*
+ Find t such that curve_point (t)[AX] == COORDINATE
+*/
+vector<Real>
+Bezier::solve_point (Axis ax, Real coordinate) const
{
- for (int i = 1; i < encompass_.size () - 1; i++)
- if (encompass_[i].y () > y (encompass_[i].x ()))
- return false;
- return true;
-}
+ Polynomial p (polynomial (ax));
+ p.coefs_[0] -= coordinate;
-Real
-Bezier_bow::check_fit_f ()
-{
- Real dy = 0;
- for (int i = 1; i < encompass_.size () - 1; i++)
- dy = dy >? (encompass_[i].y () - y (encompass_[i].x ()));
- return dy;
+ vector<Real> sol (p.solve ());
+ return filter_solutions (sol);
}
-void
-Bezier_bow::set (Array<Offset> points, int dir)
+/**
+ Compute the bounding box dimensions in direction of A.
+*/
+Interval
+Bezier::extent (Axis a) const
{
- dir_ = dir;
- encompass_ = points;
+ 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;
}
-void
-Bezier_bow::transform ()
+Interval
+Bezier::control_point_extent (Axis a) const
{
- origin_ = encompass_[0];
- encompass_.translate (-origin_);
-
- Offset delta = encompass_[encompass_.size () - 1] - encompass_[0];
- /*
- Real dx = encompass_[encompass_.size () - 1].x () - encompass_[0].x ();
- Real dy = encompass_[encompass_.size () - 1].y () - encompass_[0].y ();
- */
-
- alpha_ = delta.arg ();
- encompass_.rotate (-alpha_);
+ Interval ext;
+ for (int i = CONTROL_COUNT; i--;)
+ ext.add_point (control_[i][a]);
- if (dir_ == DOWN)
- encompass_.flipy ();
+ return ext;
}
+
+/**
+ Flip around axis A
+*/
void
-Bezier_bow::transform_controls_back ()
+Bezier::scale (Real x, Real y)
{
- // silly name; let's transform encompass back too
- // to allow recalculation without re-set()ting encompass array
- if (dir_ == DOWN)
+ for (int i = CONTROL_COUNT; i--;)
{
- control_.flipy ();
- return_.flipy ();
- encompass_.flipy ();
+ control_[i][X_AXIS] = x * control_[i][X_AXIS];
+ control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
}
+}
- control_.rotate (alpha_);
- control_.translate (origin_);
-
- return_.rotate (alpha_);
- return_.translate (origin_);
-
- encompass_.rotate (alpha_);
- encompass_.translate (origin_);
+void
+Bezier::rotate (Real phi)
+{
+ Offset rot (complex_exp (Offset (0, phi)));
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ control_[i] = complex_multiply (rot, control_[i]);
}
void
-Bezier_bow::calc_default (Real h)
+Bezier::translate (Offset o)
{
- Real pi = M_PI;
-#ifndef STANDALONE
- Real staffsize_f = paper_l_->get_var ("barsize");
-#else
- Real staffsize_f = 16 PT;
-#endif
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ control_[i] += o;
+}
- 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;
-
-
- Offset delta (encompass_[encompass_.size () - 1].x () - encompass_[0].x (), 0);
-
- Real d = 1;
-
- Real b = delta.length ();
- 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] = {0, 0, indent, height, b - indent, height, b, 0 };
- Array<Offset> control;
- control.push (Offset (0, 0));
- control.push (Offset (indent, height));
- control.push (Offset (b - indent, height));
- control.push (Offset (b, 0));
- Bezier::set (control);
-
-// Real phi = dx ? atan (dy/dx) : sign (dy) * pi / 2.0;
-// control.rotate (phi);
+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;
+}
projects / lilypond.git / blobdiff