lily/bezier-bow.cc

   1 /*
   2   bezier.cc -- implement Bezier and Bezier_bow
   3
   4   source file of the GNU LilyPond music typesetter
   5
   6   (c) 1998--2004 Jan Nieuwenhuizen <janneke@gnu.org>
   7 */
   8
   9 #include <math.h>
  10
  11 #include "bezier-bow.hh"
  12 #include "misc.hh"
  13 #include "bezier.hh"
  14
  15
  16 static Real
  17 F0_1 (Real x)
  18 {
  19   return 2 / M_PI * atan (M_PI * x / 2);
  20 }
  21
  22 Real
  23 slur_height (Real width, Real h_inf, Real r_0)
  24 {
  25   return F0_1 (width * r_0 / h_inf) * h_inf;
  26 }
  27
  28   /*
  29
  30
  31   ^              x                    x
  32   |
  33 height   <indent>
  34   |
  35   v      x                                    x
  36
  37
  38
  39   For small w, the height should be proportional to w, for w ->
  40   infinity, the height should rise to a limit asymptotically.
  41
  42   Hence we take F (x) such that
  43   F (0) = 0 , F' (0) = 1, and F (infty) = 1
  44
  45   and use
  46
  47   h = h_infinity * F (x * r_0 / h_infinity)
  48
  49
  50   Examples:
  51
  52   * F (x) = 2/pi * atan (pi x/2)
  53
  54   * F (x) = 1/alpha * x^alpha / (1 + x^alpha)
  55
  56   * (etc.)
  57
  58   [with the 2nd recipe you can determine how quickly the conversion from
  59   `small' slurs to `big' slurs occurs.]
  60
  61   Although this might seem cand_idates to SCM-ify, it is not all clear
  62   which parameters (ie. h_inf, r_0, F (.)) should be candidates for
  63   this.  At present h_inf and r_0 come from paper settings, but we did
  64   no experiments for determining the best combinations of F, h_inf and
  65   r_0.
  66
  67
  68   The indent is proportional to the height of the slur for small
  69   slurs.  For large slurs, this gives a certain hookiness at the end,
  70   so we increase the indent.
  71
  72   ind = G(w)
  73
  74   w -> 0,  G(w) -> .5 h
  75
  76   w -> inf, G(w) -> 2*h
  77
  78   eg.
  79
  80
  81   G(w) = h (w/(w+h_inf) 1.5 + .5 h
  82
  83
  84   */
  85
  86 Bezier
  87 slur_shape (Real width, Real h_inf, Real r_0)
  88 {
  89   Bezier curve;
  90   Real height =  slur_height (width, h_inf, r_0);
  91   Real indent = (width/(h_inf+ width)*1.5 + 0.5) * height;
  92
  93   curve.control_[0] = Offset (0, 0);
  94   curve.control_[1] = Offset (indent, height);
  95   curve.control_[2] = Offset (width - indent, height);
  96   curve.control_[3] = Offset (width, 0);
  97   return curve;
  98 }
  99