/*
- 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--1999 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.
+
+ 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 "bezier-bow.hh"
#include "misc.hh"
#include "bezier.hh"
-#include "dimensions.hh"
-#include "direction.hh"
-#include "paper-def.hh"
-#include "debug.hh"
-#include "main.hh"
-void
-flipy (Array<Offset> &c)
+static Real
+F0_1 (Real x)
{
- for (int i = c.size (); i--;)
- c[i][Y_AXIS] = - c[i][Y_AXIS];
+ return 2 / M_PI * atan (M_PI * x / 2);
}
-void
-rotate (Array<Offset> &c, Real phi)
+Real
+slur_height (Real width, Real h_inf, Real r_0)
{
- Offset rot (complex_exp (Offset (0, phi)));
- for (int i = 0; i < c.size (); i++)
- c[i] = complex_multiply (rot, c[i]);
+ return F0_1 (width * r_0 / h_inf) * h_inf;
}
-void
-translate (Array<Offset> &c, Offset o)
-{
- for (int i = 0; i < c.size (); i++)
- c[i] += o;
-}
+/*
+ ^ x x
+ |
+ height <indent>
+ |
+ v x x
-Bezier_bow::Bezier_bow (Paper_def* paper_l)
-{
- paper_l_ = paper_l;
-}
-void
-Bezier_bow::blow_fit ()
-{
- Real x1 = encompass_[0][X_AXIS];
- Real x2 = encompass_.top ()[X_AXIS];
-
- Real factor = 1.0;
- for (int i=1; i < encompass_.size ()-1; i++)
- {
- if (encompass_[i][X_AXIS] > x1 && encompass_[i][X_AXIS] < x2)
- {
- Real y = curve_.get_other_coordinate (X_AXIS, encompass_[i][X_AXIS]);
- if (y>0)
- {
- Real f = encompass_[i][Y_AXIS] / y;
- factor = factor >? f;
- }
- }
- }
-
- curve_.control_[1][Y_AXIS] *= factor;
- curve_.control_[2][Y_AXIS] *= factor;
- return_.control_[1][Y_AXIS] *= factor;
- return_.control_[2][Y_AXIS] *= factor;
-
- curve_.check_sanity ();
-}
+ For small w, the height should be proportional to w, for w ->
+ infinity, the height should rise to a limit asymptotically.
-Real
-Bezier_bow::calc_f (Real height)
-{
- transform ();
- calc_default (height);
+ Hence we take F (x) such that
+ F (0) = 0 , F' (0) = 1, and F (infty) = 1
- Real dy = check_fit_f ();
- calc_return (0, 0);
+ and use
- transform_back ();
- return dy;
-}
+ h = h_infinity * F (x * r_0 / h_infinity)
-void
-Bezier_bow::calc ()
-{
- transform ();
- calc_controls ();
- transform_back ();
-}
+ Examples:
+ * F (x) = 2/pi * atan (pi x/2)
-/*
- [TODO]
- * see if it works
- * document in Documentation/fonts.tex
- */
+ * F (x) = 1/alpha * x^alpha / (1 + x^alpha)
-/*
- 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 ()
-{
- 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");
-
- Real b = curve_.control_[3][X_AXIS] - curve_.control_[0][X_AXIS];
- Real clip_h = clip_ratio * b <? clip_height;
- Real begin_h = curve_.control_[1][Y_AXIS] - curve_.control_[0][Y_AXIS];
- Real end_h = curve_.control_[2][Y_AXIS] - curve_.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 = (curve_.control_[1] - curve_.control_[0]).arg () + dir_ * alpha_;
- Real end_alpha = pi - (curve_.control_[2] - curve_.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_.top ()[Y_AXIS] += dir_ * dy;
- }
- else
- {
- //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_.top ()[Y_AXIS] += dir_ * end_dy;
-
- Offset delta = encompass_.top () - encompass_[0];
- alpha_ = delta.arg ();
- }
-
- transform ();
-
- return again;
-}
+ * (etc.)
-void
-Bezier_bow::calc_controls ()
-{
- for (int i = 0; i < 3; i++)
- {
-
- if (i && !calc_clipping ())
- return;
-
- /*
- why do we always recalc from 0?
- shouldn't calc_f () be used (too), rather than blow_fit () (only)?
- */
- calc_default (0);
- curve_.check_sanity ();
- if (check_fit_f () > 0)
- {
- calc_tangent_controls ();
- blow_fit ();
- }
- else
- {
- calc_return (0, 0);
- return;
- }
- }
-}
+ [with the 2nd recipe you can determine how quickly the conversion from
+ `small' slurs to `big' slurs occurs.]
-void
-Bezier_bow::calc_return (Real begin_alpha, Real end_alpha)
-{
- Real thick = paper_l_->get_var ("slur_thickness");
+ Although this might seem cand_idates to SCM-ify, it is not all clear
+ which parameters (ie. h_inf, r_0, F (.)) should be candidates for
+ this. At present h_inf and r_0 come from layout settings, but we did
+ no experiments for determining the best combinations of F, h_inf and
+ r_0.
- return_.control_[0] = curve_.control_[3];
- return_.control_[3] = curve_.control_[0];
- return_.control_[1] = curve_.control_[2] - thick * complex_exp (Offset (0, 90 + end_alpha));
- return_.control_[2] = curve_.control_[1] - thick * complex_exp (Offset (0, 90 - begin_alpha));
-}
+ The indent is proportional to the height of the slur for small
+ slurs. For large slurs, this gives a certain hookiness at the end,
+ so we increase the indent.
-/*
-This function calculates 2 center control points, based on
-
- See Documentation/fonts.tex
- */
-void
-Bezier_bow::calc_tangent_controls ()
-{
- Real b = curve_.control_[3][X_AXIS];
- Real h = curve_.control_[1][Y_AXIS];
-
-
- Drul_array<Offset> disturb;
- Drul_array<Real> maxtan;
- maxtan[LEFT] = maxtan[RIGHT] = h/(b/2);
- disturb[LEFT] = disturb[RIGHT] = Offset (b / 2, h);
-
- for (int i = 1; i < encompass_.size () -1; i++)
- {
- Real y= encompass_[i][Y_AXIS];
- if (y> 0)
- {
- Real x = encompass_[i][X_AXIS];
-
- Direction d = LEFT;
- do
- {
- // 1 if d == LEFT
- int k = (1 - d)/2;
- Real tan = y / ((1-k)* b - d * x);
-
- if (tan > maxtan[d])
- {
- maxtan[d] = tan;
- disturb[d] = Offset (x,y);
- }
- }
- while (flip (&d)!=LEFT);
- }
- }
-
- for (int i = 0; i < encompass_.size (); i++ )
- h = h >? encompass_[i][Y_AXIS];
-
- /*
- The curve will always be under line between curve_.control_0 -> curve_.control_1, so
- make it extra steep by slur_rc_factor
- */
- Real rc_correct = paper_l_->get_var ("slur_rc_factor");
-
- Drul_array<Real> angles;
- Direction d = LEFT;
- do
- {
- maxtan[d] *= rc_correct;
- angles[d] = atan (-d * maxtan[d]);
- }
- while (flip(&d) != LEFT);
-
- Real rc3 = 0.0;
-
- // if we have two disturbing points, have line through those...
- if (disturb[LEFT][Y_AXIS] != disturb[RIGHT][Y_AXIS])
- rc3 = (disturb[RIGHT][Y_AXIS] - disturb[LEFT][Y_AXIS]) / (disturb[RIGHT][X_AXIS] - disturb[LEFT][X_AXIS]);
-
- else
- rc3 = tan ((angles[RIGHT] - angles[LEFT]) / 2);
-
-
- // ugh: be less steep
- rc3 /= 2*rc_correct;
-
-
- Real c2 = -maxtan[RIGHT] * curve_.control_[3][X_AXIS];
-
- // use highest because rc3 is damped.
- Real maxy = disturb[LEFT][Y_AXIS] >? disturb[RIGHT][Y_AXIS];
- Real c3 = disturb[LEFT][Y_AXIS] > disturb[RIGHT][Y_AXIS] ?
- maxy - rc3 * disturb[LEFT][X_AXIS] :
- maxy - rc3 * disturb[RIGHT][X_AXIS];
-
- curve_.control_[1][X_AXIS] = c3 / (maxtan[LEFT] - rc3);
- curve_.control_[1][Y_AXIS] = maxtan[LEFT] * curve_.control_[1][X_AXIS];
-
- curve_.control_[2][X_AXIS] = (c3 - c2) / (maxtan[RIGHT] - rc3);
- curve_.control_[2][Y_AXIS] = maxtan[RIGHT] * curve_.control_[2][X_AXIS] + c2;
-
-
- curve_.check_sanity();
-
- calc_return (angles[LEFT], angles[RIGHT]);
-}
+ indent = G (w)
-/*
- The maximum amount that the encompass points stick out above the bezier curve.
- */
-Real
-Bezier_bow::check_fit_f () const
-{
- Real dy = 0;
- Real x1 = encompass_[0][X_AXIS];
- Real x2 = encompass_.top ()[X_AXIS];
- for (int i = 1; i < encompass_.size () - 1; i++)
- {
- Real x = encompass_[i][X_AXIS];
- if (x1< x&& x < x2)
- dy = dy >? (encompass_[i][Y_AXIS] - curve_.get_other_coordinate (X_AXIS, x));
- }
- return dy;
-}
+ w -> 0, G (w) -> .33 w
-void
-Bezier_bow::set (Array<Offset> points, Direction dir)
-{
- dir_ = dir;
- encompass_ = points;
-}
+ (due to derivative constraints, we cannot have indent > len/3)
-void
-Bezier_bow::transform ()
-{
- origin_ = encompass_[0];
- translate (encompass_,-origin_);
+ w -> inf, G (w) -> 2*h_inf
- Offset delta = encompass_.top () - encompass_[0];
- alpha_ = delta.arg ();
+ i.e.
- rotate (encompass_, -alpha_);
- if (dir_ == DOWN)
- flipy (encompass_);
-}
+ G (0) = 0 , G'(0) 1/3, G (infty) = 2h_inf
-void
-Bezier_bow::transform_back ()
-{
- if (dir_ == DOWN)
- {
- curve_.flip (Y_AXIS);
- return_.flip (Y_AXIS);
- flipy (encompass_);
- }
-
- curve_.rotate (alpha_);
- curve_.translate (origin_);
-
- return_.rotate (alpha_);
- return_.translate (origin_);
-
- rotate (encompass_,alpha_);
- translate (encompass_,origin_);
-}
+ solve from
-/*
- See Documentation/fonts.tex
- */
-void
-Bezier_bow::calc_default (Real h)
-{
- Real pi = M_PI;
+ G (w) = r + p/(w+q)
- Real height_limit = paper_l_->get_var ("slur_height_limit");
- Real ratio = paper_l_->get_var ("slur_ratio");
+ yields
- Real alpha = height_limit * 2.0 / pi;
- Real beta = pi * ratio / (2.0 * height_limit);
+ G (w) = 2 h_inf - max_fraction * q^2/ (w + q)
- Offset delta (encompass_.top ()[X_AXIS]
- - encompass_[0][X_AXIS], 0);
+ with q = 2 h_inf
+*/
+
+void
+get_slur_indent_height (Real *indent, Real *height,
+ Real width, Real h_inf, Real r_0)
+{
+ Real max_fraction = 1.0 / 3.1;
+ *height = slur_height (width, h_inf, r_0);
- Real b = delta.length ();
- Real indent = alpha * atan (beta * b);
- Real height = indent + h;
-
- curve_.control_ [0] = Offset (0, 0);
- curve_.control_ [1] = Offset (indent, height);
- curve_.control_ [2] = Offset (b - indent, height);
- curve_.control_ [3] = Offset (b, 0);
+ Real q = 2 * h_inf / max_fraction;
+ *indent = 2 * h_inf - sqr (q) * max_fraction / (width + q);
}
+Bezier
+slur_shape (Real width, Real h_inf, Real r_0)
+{
+ Real indent;
+ Real height;
+
+ get_slur_indent_height (&indent, &height,
+ width, h_inf, r_0);
+ Bezier curve;
+ curve.control_[0] = Offset (0, 0);
+ curve.control_[1] = Offset (indent, height);
+ curve.control_[2] = Offset (width - indent, height);
+ curve.control_[3] = Offset (width, 0);
+ return curve;
+}
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