source file of the GNU LilyPond music typesetter
- (c) 1998--1999 Jan Nieuwenhuizen <janneke@gnu.org>
+ (c) 1998--2008 Jan Nieuwenhuizen <janneke@gnu.org>
*/
-#include <math.h>
#include "bezier.hh"
-#include "misc.hh"
-
-#ifndef STANDALONE
-#include "dimensions.hh"
-#include "direction.hh"
-#include "paper-def.hh"
-#include "debug.hh"
-#include "main.hh"
-#define BEZIER_BOW_DOUT if (check_debug && !monitor->silent_b ("Bezier_bow")) cout
-#else
-#define BEZIER_BOW_DOUT cerr
-#endif
+#include "warn.hh"
+#include "libc-extension.hh"
-void
-Curve::flipy ()
-{
- 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_AXIS] > 0)
- {
- Real phi = (*this)[i][Y_AXIS] / (*this)[i][X_AXIS];
- 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 = 0; i < size (); i++)
- (*this)[i] = complex_multiply (rot, (*this)[i]);
+ 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 = 0; i < size (); i++)
- (*this)[i] += o;
+ for (vsize i = 0; i < array->size (); i++)
+ (*array)[i] += o;
}
-Bezier::Bezier ()
-{
- control_.set_size (4);
-}
+/*
+ Formula of the bezier 3-spline
-void
-Bezier::calc (int steps)
-{
- steps = steps >? 10;
- curve_.set_size (steps);
- 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::print () const
-{
-#ifndef NPRINT
- if (check_debug && !monitor->silent_b ("Bezier_controls"))
- {
- if (control_[1].length ())
- {
- cout << "Bezier\n";
- cout << "Controls: ";
- for (int i=0; i < control_.size (); i++)
- cout << control_[i].str () << ", ";
-// cout << "\n";
- }
- }
-#endif
-}
-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
{
- // ugh
- // bounds func should be templatised to take array of offsets too?
- Array<Real> positions;
- for (int i = 0; i < curve_.size (); i++)
- positions.push (curve_[i][X_AXIS]);
-
- Slice slice = get_bounds_slice (positions, x);
- // ugh
- Offset z1 = curve_[0 >? slice[BIGGER] - 1];
- Offset z2 = curve_[1 >? slice[BIGGER]];
- Real multiplier = (x - z2[X_AXIS]) / (z1[X_AXIS] - z2[X_AXIS]);
- Real y = z1[Y_AXIS] * multiplier + (1.0 - multiplier) * z2[Y_AXIS];
-
- return y;
-}
-
+ Axis other = Axis ((a +1) % NO_AXES);
+ vector<Real> ts = solve_point (a, x);
-Bezier_bow::Bezier_bow (Paper_def* paper_l)
-{
- paper_l_ = paper_l;
- return_.set_size (4);
-}
+ if (ts.size () == 0)
+ {
+ programming_error ("no solution found for Bezier intersection");
+ return 0.0;
+ }
-void
-Bezier_bow::blow_fit ()
-{
- Real dy1 = check_fit_f ();
- if (!dy1)
- return;
-
- // be careful not to take too big step
- Real f = 0.3;
- Real h1 = dy1 * f;
- control_[1][Y_AXIS] += h1;
- control_[2][Y_AXIS] += h1;
- return_[1][Y_AXIS] += h1;
- return_[2][Y_AXIS] += h1;
-
- calc_bezier ();
- Real dy2 = check_fit_f ();
- if (!dy2)
- return;
-
-#ifndef STANDALONE
- Real internote_f = paper_l_->get_realvar (interline_scm_sym)/2.0;
-#else
- Real internote_f = STAFFHEIGHT / 8;
+#ifdef PARANOID
+ Offset c = curve_point (ts[0]);
+ if (fabs (c[a] - x) > 1e-8)
+ programming_error ("bezier intersection not correct?");
#endif
- Real epsilon = internote_f / 4;
- if (abs (dy2 - dy1) < epsilon)
- return;
-
- /*
- Assume
- dy = B (h)
- with
- B (h) = a * h + b;
-
- Then we get for h : B (h) = 0
-
- B(0) = dy1 = a * 0 + b => b = dy1
- B(h1) = dy2 = a * h1 + b => a * f * dy1 + b = dy2
-
- =>
-
- a * dy1 / 2 + dy1 = dy2 => a = (dy2 - dy1) / (f * dy1)
- */
-
- Real a = (dy2 - dy1) / (f * dy1);
- Real b = dy1;
- Real h = -b / a;
-
- if (sign (h) != sign (h1))
- return;
-
- if (sign (h) != sign (h1))
- return;
-
- control_[1][Y_AXIS] += -h1 +h;
- control_[2][Y_AXIS] += -h1 +h;
- return_[1][Y_AXIS] += -h1 +h;
- return_[2][Y_AXIS] += -h1 +h;
-}
-
-void
-Bezier_bow::calc_bezier ()
-{
- Real s = sqrt (control_[3][X_AXIS] * control_[3][X_AXIS]
- + control_[1][Y_AXIS] * control_[2][Y_AXIS]);
-#ifndef STANDALONE
- Real internote = paper_l_->get_realvar (interline_scm_sym)/2.0;
-#else
- Real internote = STAFFHEIGHT / 8;
-#endif
- int steps = (int)rint (s / internote);
- Bezier::calc (steps);
+ return curve_coordinate (ts[0], other);
}
Real
-Bezier_bow::calc_f (Real height)
+Bezier::curve_coordinate (Real t, Axis a) const
{
- transform ();
- calc_default (height);
-
- calc_bezier ();
+ 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];
- Real dy = check_fit_f ();
- calc_return (0, 0);
+ tj *= t;
+ }
- transform_back ();
- return dy;
+ return r;
}
-void
-Bezier_bow::calc ()
+Offset
+Bezier::curve_point (Real t) const
{
-#ifndef NPRINT
-// if (check_debug && !monitor->silent_b ("Bezier_bow_controls"))
- if (check_debug && !(monitor->silent_b ("Bezier_controls")
- && monitor->silent_b ("Bezier_bow_controls")))
+ 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++)
{
- cout << "Before transform*********\n";
- print ();
- cout << "************************\n";
- }
-#endif
- transform ();
- print ();
-
- calc_controls ();
+ o += control_[j] * binomial_coefficient_3[j]
+ * tj * one_min_tj[3 - j];
- print ();
- transform_back ();
-#ifndef NPRINT
-// if (check_debug && !monitor->silent_b ("Bezier_bow_controls"))
- if (check_debug && !(monitor->silent_b ("Bezier_controls")
- && monitor->silent_b ("Bezier_bow_controls")))
- {
- cout << "After transform*********\n";
- print ();
- cout << "************************\n";
+ tj *= 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);
#endif
-}
-/*
- [TODO]
- * see if it works
- * document in Documentation/fonts.tex
- */
+ return o;
+}
/*
- Clipping
-
- This function tries to address two issues:
- * the tangents of the slur should always point inwards
- in the actual slur, i.e. *after rotating back*.
-
- * slurs shouldn't be too high
- let's try : h <= 1.2 b && h <= 3 staffheight?
-
- We could calculate the tangent of the bezier curve from
- both ends going inward, and clip the slur at the point
- where the tangent (after rotation) points up (or inward
- with a certain maximum angle).
-
- However, we assume that real clipping is not the best
- answer. We expect that moving the outer control point up
- if the slur becomes too high will result in a nicer slur
- after recalculation.
-
- Knowing that the tangent is the line through the first
- two control points, we'll clip (move the outer control
- point upwards) too if the tangent points outwards.
- */
-
-bool
-Bezier_bow::calc_clipping ()
+ 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
{
-#ifndef STANDALONE
- Real clip_height = paper_l_->get_var ("slur_clip_height");
- Real clip_ratio = paper_l_->get_var ("slur_clip_ratio");
- Real clip_angle = paper_l_->get_var ("slur_clip_angle");
-#else
- Real staffsize_f = STAFFHEIGHT;
- Real clip_height = 3.0 * staffsize_f;
- Real clip_ratio = 1.2;
- Real clip_angle = 100;
-#endif
-
- Real b = control_[3][X_AXIS] - control_[0][X_AXIS];
- Real clip_h = clip_ratio * b <? clip_height;
- Real begin_h = control_[1][Y_AXIS] - control_[0][Y_AXIS];
- Real end_h = control_[2][Y_AXIS] - control_[3][Y_AXIS];
- Real begin_dy = 0 >? begin_h - clip_h;
- Real end_dy = 0 >? end_h - clip_h;
-
- Real pi = M_PI;
- Real begin_alpha = (control_[1] - control_[0]).arg () + dir_ * alpha_;
- Real end_alpha = pi - (control_[2] - control_[3]).arg () - dir_ * alpha_;
-
- Real max_alpha = clip_angle / 90 * pi / 2;
- if ((begin_dy < 0) && (end_dy < 0)
- && (begin_alpha < max_alpha) && (end_alpha < max_alpha))
- return false;
-
- transform_back ();
-
- bool again = true;
-
- if ((begin_dy > 0) || (end_dy > 0))
- {
- Real dy = (begin_dy + end_dy) / 4;
- dy *= cos (alpha_);
- encompass_[0][Y_AXIS] += dir_ * dy;
- encompass_[encompass_.size () - 1][Y_AXIS] += dir_ * dy;
- }
- else
+ Polynomial p (0.0);
+ Polynomial q;
+ for (int j = 0; j <= 3; j++)
{
- //ugh
- Real c = 0.4;
- if (begin_alpha >= max_alpha)
- begin_dy = 0 >? c * begin_alpha / max_alpha * begin_h;
- if (end_alpha >= max_alpha)
- end_dy = 0 >? c * end_alpha / max_alpha * end_h;
-
- encompass_[0][Y_AXIS] += dir_ * begin_dy;
- encompass_[encompass_.size () - 1][Y_AXIS] += dir_ * end_dy;
-
- Offset delta = encompass_[encompass_.size () - 1] - encompass_[0];
- alpha_ = delta.arg ();
+ q = poly_cache.terms_[j];
+ q *= control_[j][a];
+ p += q;
}
- transform ();
-
- return again;
+ return p;
}
-void
-Bezier_bow::calc_controls ()
+/**
+ Remove all numbers outside [0, 1] from SOL
+*/
+vector<Real>
+filter_solutions (vector<Real> sol)
{
- for (int i = 0; i < 3; i++)
- {
- if (i && !calc_clipping ())
- return;
-
- calc_default (0);
- calc_bezier ();
-
- if (check_fit_bo ())
- {
- calc_return (0, 0);
- return;
- }
- calc_tangent_controls ();
-
- blow_fit ();
- // ugh
- blow_fit ();
- }
+ for (vsize i = sol.size (); i--;)
+ if (sol[i] < 0 || sol[i] > 1)
+ sol.erase (sol.begin () + i);
+ return sol;
}
-void
-Bezier_bow::calc_return (Real begin_alpha, Real end_alpha)
+/**
+ find t such that derivative is proportional to DERIV
+*/
+vector<Real>
+Bezier::solve_derivative (Offset deriv) const
{
-#ifndef STANDALONE
- Real thick = paper_l_->get_var ("slur_thickness");
-#else
- Real thick = 1.8 * 0.4 PT;
-#endif
+ Polynomial xp = polynomial (X_AXIS);
+ Polynomial yp = polynomial (Y_AXIS);
+ xp.differentiate ();
+ yp.differentiate ();
- return_[0] = control_[3];
- return_[3] = control_[0];
+ Polynomial combine = xp * deriv[Y_AXIS] - yp * deriv [X_AXIS];
- 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 filter_solutions (combine.solve ());
}
/*
- See Documentation/fonts.tex
- */
-void
-Bezier_bow::calc_tangent_controls ()
+ Find t such that curve_point (t)[AX] == COORDINATE
+*/
+vector<Real>
+Bezier::solve_point (Axis ax, Real coordinate) const
{
- Offset ijk_p (control_[3][X_AXIS] / 2, control_[1][Y_AXIS]);
- BEZIER_BOW_DOUT << "ijk: " << ijk_p[X_AXIS] << ", " << ijk_p[Y_AXIS] << endl;
-
- Real default_rc = ijk_p[Y_AXIS] / ijk_p[X_AXIS];
-
- int begin_disturb = encompass_.largest_disturbing ();
- Offset begin_p = begin_disturb ? Offset (encompass_[begin_disturb][X_AXIS],
- encompass_[begin_disturb][Y_AXIS]) : ijk_p;
- Real begin_rc = begin_p[Y_AXIS] / begin_p[X_AXIS];
- if (default_rc > begin_rc)
- {
- begin_p = ijk_p;
- begin_rc = default_rc;
- }
+ Polynomial p (polynomial (ax));
+ p.coefs_[0] -= coordinate;
- Curve reversed;
- reversed.set_size (encompass_.size ());
- Real b = control_[3][X_AXIS];
- for (int i = 0; i < encompass_.size (); i++ )
- {
- // b 1 0
- // r = - * c
- // 0 0 -1
- reversed[i][X_AXIS] = b - encompass_[encompass_.size () - i - 1][X_AXIS];
- reversed[i][Y_AXIS] = encompass_[encompass_.size () - i - 1][Y_AXIS];
- }
+ vector<Real> sol (p.solve ());
+ return filter_solutions (sol);
+}
- 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_AXIS],
- encompass_[end_disturb][Y_AXIS]) : ijk_p;
- Real end_rc = end_p[Y_AXIS] / (control_[3][X_AXIS] - end_p[X_AXIS]);
- if (default_rc > end_rc)
+/**
+ 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--;)
{
- end_p = ijk_p;
- end_rc = default_rc;
+ Offset o (curve_point (sols[i]));
+ iv.unite (Interval (o[a], o[a]));
}
- BEZIER_BOW_DOUT << "begin " << begin_p[X_AXIS] << ", " << begin_p[Y_AXIS] << endl;
- BEZIER_BOW_DOUT << "end " << end_p[X_AXIS] << ", " << end_p[Y_AXIS] << endl;
-
- Real height =control_[1][Y_AXIS];
- for (int i = 0; i < encompass_.size (); i++ )
- height = height >? encompass_[i][Y_AXIS];
-
- // emperic computer science:
- // * tangents somewhat steeper than minimal line
-#ifndef STANDALONE
- Real internote = paper_l_->get_realvar (interline_scm_sym)/2.0;
- Real rc_correct = paper_l_->get_var ("slur_rc_factor");
-#else
- Real internote = STAFFHEIGHT / 8;
- Real rc_correct = 2.4;
-#endif
-
- 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;
-
- Real epsilon = internote / 5;
-
- // if we have two disturbing points, have height line through those...
- if (!((abs (begin_p[X_AXIS] - end_p[X_AXIS]) < epsilon)
- && (abs (begin_p[Y_AXIS] - end_p[Y_AXIS]) < epsilon)))
- theta = atan (end_p[Y_AXIS] - begin_p[Y_AXIS]) / (end_p[X_AXIS] - begin_p[X_AXIS]);
-
- Real rc3 = tan (theta);
- // ugh: be less steep
- rc3 /= 2*rc_correct;
-
- Real c2 = -rc2 * control_[3][X_AXIS];
- Real c3 = begin_p[Y_AXIS] > end_p[Y_AXIS] ? begin_p[Y_AXIS]
- - rc3 * begin_p[X_AXIS] : end_p[Y_AXIS] - rc3 * end_p[X_AXIS];
-
- BEZIER_BOW_DOUT << "y1 = " << rc1 << " x + 0" << endl;
- BEZIER_BOW_DOUT << "y2 = " << rc2 << " x + " << c2 << endl;
- BEZIER_BOW_DOUT << "y3 = " << rc3 << " x + " << c3 << endl;
- control_[1][X_AXIS] = c3 / (rc1 - rc3);
- control_[1][Y_AXIS] = rc1 * control_[1][X_AXIS];
- control_[2][X_AXIS] = (c3 - c2) / (rc2 - rc3);
- BEZIER_BOW_DOUT << "c2[X_AXIS] = " << control_[2][X_AXIS] << endl;
- BEZIER_BOW_DOUT << "(c3 - c2) = " << (c3 - c2) << endl;
- BEZIER_BOW_DOUT << "(rc2 - rc3) = " << (rc2 - rc3) << endl;
- control_[2][Y_AXIS] = rc2 * control_[2][X_AXIS] + c2;
- BEZIER_BOW_DOUT << "c2[Y_AXIS]" << control_[2][Y_AXIS] << endl;
-
- calc_return (begin_alpha, end_alpha);
+ return iv;
}
-bool
-Bezier_bow::check_fit_bo ()
+Interval
+Bezier::control_point_extent (Axis a) const
{
- for (int i = 1; i < encompass_.size () - 1; i++)
- if ((encompass_[i][X_AXIS] > encompass_[0][X_AXIS])
- && (encompass_[i][X_AXIS] < encompass_[encompass_.size () -1][X_AXIS]))
- if (encompass_[i][Y_AXIS] > y (encompass_[i][X_AXIS]))
- return false;
- return true;
-}
+ Interval ext;
+ for (int i = CONTROL_COUNT; i--;)
+ ext.add_point (control_[i][a]);
-Real
-Bezier_bow::check_fit_f ()
-{
- Real dy = 0;
- for (int i = 1; i < encompass_.size () - 1; i++)
- if ((encompass_[i][X_AXIS] > encompass_[0][X_AXIS])
- && (encompass_[i][X_AXIS] < encompass_[encompass_.size () -1][X_AXIS]))
- dy = dy >? (encompass_[i][Y_AXIS] - y (encompass_[i][X_AXIS]));
- return dy;
+ return ext;
}
+
+/**
+ Flip around axis A
+*/
void
-Bezier_bow::print () const
+Bezier::scale (Real x, Real y)
{
-#ifndef NPRINT
- Bezier::print ();
- if (check_debug && !monitor->silent_b ("Bezier_bow_controls"))
+ for (int i = CONTROL_COUNT; i--;)
{
- cout << "Bezier_bow\n";
- cout << "Encompass: ";
- for (int i=0; i < encompass_.size (); i++)
- cout << encompass_[i].str () << ", ";
-// cout << "\n";
+ control_[i][X_AXIS] = x * control_[i][X_AXIS];
+ control_[i][Y_AXIS] = y * control_[i][Y_AXIS];
}
-#endif
}
void
-Bezier_bow::set (Array<Offset> points, int dir)
+Bezier::rotate (Real phi)
{
- dir_ = dir;
- encompass_ = points;
+ 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::transform ()
+Bezier::translate (Offset o)
{
- origin_ = encompass_[0];
- encompass_.translate (-origin_);
-
- Offset delta = encompass_[encompass_.size () - 1] - encompass_[0];
- alpha_ = delta.arg ();
-
- encompass_.rotate (-alpha_);
-
- if (dir_ == DOWN)
- encompass_.flipy ();
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ control_[i] += o;
}
void
-Bezier_bow::transform_back ()
+Bezier::assert_sanity () const
{
- if (dir_ == DOWN)
- {
- control_.flipy ();
- return_.flipy ();
- encompass_.flipy ();
- }
-
- control_.rotate (alpha_);
- control_.translate (origin_);
-
- return_.rotate (alpha_);
- return_.translate (origin_);
-
- encompass_.rotate (alpha_);
- encompass_.translate (origin_);
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ assert (!isnan (control_[i].length ())
+ && !isinf (control_[i].length ()));
}
-/*
- See Documentation/fonts.tex
- */
void
-Bezier_bow::calc_default (Real h)
+Bezier::reverse ()
{
- Real pi = M_PI;
-#ifndef STANDALONE
- Real height_limit = paper_l_->get_var ("slur_height_limit");
- Real ratio = paper_l_->get_var ("slur_ratio");
-#else
- Real staffsize_f = STAFFHEIGHT;
- Real height_limit = staffsize_f;
- Real ratio = 1.0/3.0;
-#endif
-
- Real alpha = height_limit * 2.0 / pi;
- Real beta = pi * ratio / (2.0 * height_limit);
-
- Offset delta (encompass_[encompass_.size () - 1][X_AXIS]
- - encompass_[0][X_AXIS], 0);
- Real b = delta.length ();
- Real indent = alpha * atan (beta * b);
- Real height = indent + h;
-
- 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);
+ Bezier b2;
+ for (int i = 0; i < CONTROL_COUNT; i++)
+ b2.control_[CONTROL_COUNT - i - 1] = control_[i];
+ *this = b2;
}
-
projects / lilypond.git / blobdiff